First Edition, 2012
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[email protected] Table of Contents Chapter 1 - Symmetry Chapter 2 - Group (Mathematics) Chapter 3 - Group Action Chapter 4 - Regular Polytope Chapter 5 - Lie Point Symmetry
Chapter 1
Symmetry
Sphere symmetrical group o.
Leonardo da Vinci's Vitruvian Man (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe. Symmetry (from the Greek: "συμμετρεῖν" = to measure together), generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise. Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.
The "precise" notions of symmetry have various measures and operational definitions. For example, symmetry may be observed:
with respect to the passage of time; as a spatial relationship; through geometric transformations such as scaling, reflection, and rotation; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.
Here we, describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics as a whole. The third perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a fourth perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion. The opposite of symmetry is asymmetry.
Symmetry in geometry The most familiar type of symmetry for many people is geometrical symmetry. Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.
Reflection symmetry Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, for the same reason.
If the letter T is reflected along a vertical axis, it appears the same. Note that this is sometimes called horizontal symmetry, and sometimes vertical symmetry. One can better use an unambiguous formulation, e.g. "T has a vertical symmetry axis" or "T has leftright symmetry." The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the kites and the isosceles trapezoids. For each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space. Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane. In certain contexts there is rotational symmetry anyway. Then mirror-image symmetry is equivalent with inversion symmetry; in such contexts in modern physics the term Psymmetry is used for both (P stands for parity). For more general types of reflection there are corresponding more general types of reflection symmetry. Examples:
with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc.). with respect to circle inversion
Rotational symmetry Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m). Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws. For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m×m orthogonal matrices with determinant 1. For m=3 this is the rotation group. In another meaning of the word, the rotation group of an object is the symmetry group within E+(n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law.
Translational symmetry Translational symmetry leaves an object invariant under a discrete or continuous group of translations Ta(p) = p + a.
Glide reflection symmetry A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.
Rotoreflection symmetry In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. As symmetry groups with regard to a roto-reflection we can distinguish:
the angle has no common divisor with 360°, the symmetry group is not discrete 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n = 1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion. Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination.
Helical symmetry
A drill bit with helical symmetry. Helical symmetry is the kind of symmetry seen in such everyday objects as springs, Slinky toys, drill bits, and augers. It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis). The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at an even angular speed while simultaneously moving at another even speed along its axis of rotation (translation). At any one point in time, these two motions combine to give a coiling angle that helps define the properties of the tracing. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°.
Conversely, if the rotation is slow and the translation is speedy, the coiling angle will approach 90°. Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:
Infinite helical symmetry. If there are no distinguishing features along the length of a helix or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a cross section of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
n-fold helical symmetry. If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called n-fold helical symmetry, where n = 360°/θ. This concept can be further generalized to include cases where mθ is a multiple of 360°—that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
Non-repeating helical symmetry. This is the case in which the angle of rotation θ required to observe the symmetry is irrational. The angle of rotation never repeats exactly no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating point group in two dimensions. DNA is an example of this type of non-repeating helical symmetry.
Non-isometric symmetries A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
The group of similarity transformations, i.e. affine transformations represented by a matrix A that is a scalar times an orthogonal matrix. Thus dilations are added, self-similarity is considered a symmetry.
The group of affine transformations represented by a matrix A with determinant 1 or −1, i.e. the transformations which preserve area; this adds e.g. oblique reflection symmetry.
The group of all bijective affine transformations.
The group of Möbius transformations which preserve cross-ratios.
In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines Euclidean geometry, whereas the group of Möbius transformations defines projective geometry.
Scale symmetry and fractals Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new object has the same properties as the original. Scale symmetry is notable for the fact that it does not exist for most physical systems, a point that was first discerned by Galileo. Simple examples of the lack of scale symmetry in the physical world include the difference in the strength and size of the legs of elephants versus mice, and the observation that if a candle made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under its own weight. A more subtle form of scale symmetry is demonstrated by fractals. As conceived by Benoît Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks similar or even exactly the same no matter what degree of magnification is used to examine it. A coast is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables children to use small twigs as stand-ins for full trees in dioramas, is another example. This similarity to naturally occurring phenomena provides fractals with an everyday familiarity not typically seen with mathematically generated functions. As a consequence, they can produce strikingly beautiful results such as the Mandelbrot set. Intriguingly, fractals have also found a place in CG, or computer-generated movie effects, where their
ability to create very complex curves with fractal symmetries results in more realistic virtual worlds.
Symmetry in mathematics In formal terms, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).
Mathematical model for symmetry The set of all symmetry operations considered, on all objects in a set X, can be modeled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetrical to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric. A general example is that G is a group of bijections g: V → V acting on the set of functions x: V → W by (gx)(v) = x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v) = x(g(v)) for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v) = w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself. In a modified version for vector fields, we have (gx)(v) = h(g, x(g−1(v))) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g. The symmetry group of x consists of all g for which x(v) = h(g, x(g(v))) for all v. In this case the symmetry group of a constant function may be a proper subgroup of G: a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero. For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. The rotation group of an object is the symmetry group if G is restricted to E+(n), the group of direct isometries. Objects can be modeled as functions x, of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (x is just a Boolean function of position v), or, at the other extreme, e.g. symmetry of right and left hand with all their structure. For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same
properties, the equivalence classes are the orbits of the group action on the space itself. We need the value of x at one point in every orbit to define the full object. A set of such representatives forms a fundamental domain. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry. An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object x we can e.g.:
take the values in a fundamental domain (i.e., add copies of the object)
take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap)
If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric. As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns"). In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by angular momentum and velocity, respectively. A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of {...,1,2,5,6,9,10,13,14,...} acts transitively on all these points, while {...,1,2,3,5,6,7,9,10,11,13,14,15,...} does not act transitively on all points. Equivalently, the first set is only one conjugacy class with respect to isometries, while the second has two classes.
Symmetric functions A symmetric function is a function which is unchanged by any permutation of its variables. For example, x + y + z and xy + yz + xz are symmetric functions, whereas x2 – yz is not. A function may be unchanged by a sub-group of all the permutations of its variables. For example, ac + 3ab + bc is unchanged if a and b are exchanged; its symmetry group is isomorphic to C2.
Symmetry in logic A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, “is the same age as” is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Symmetric binary logical connectives are "and" (∧, , or &), "or" (∨), "biconditional" (if and only if) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or").
Symmetry in science Symmetry in physics Symmetry in physics has been generalized to mean invariance—that is, lack of any visible change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that "it is only slightly overstating the case to say that physics is the study of symmetry."
Symmetry in physical objects Classical objects Although an everyday object may appear exactly the same after a symmetry operation such as a rotation or an exchange of two identical parts has been performed on it, it is readily apparent that such a symmetry is true only as an approximation for any ordinary physical object. For example, if one rotates a precisely machined aluminum equilateral triangle 120 degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. However, the reality is that each corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as
optical or electron microscopes will not be fooled; he will immediately recognize that the object has been rotated by looking for details such as crystals or minor deformities. Such simple thought experiments show that assertions of symmetry in everyday physical objects are always a matter of approximate similarity rather than of precise mathematical sameness. The most important consequence of this approximate nature of symmetries in everyday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper symmetries of space and time play a major role in classical physics—that is, the physics of large, everyday objects. Quantum objects Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of quantum physics, which for the most part is the physics of very small, very simple objects such as electrons, protons, light, and atoms. Unlike everyday objects, objects such as electrons have very limited numbers of configurations, called states, in which they can exist. This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an observer is. Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion F(x) = x ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world. Consequences of quantum symmetry While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference. However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by Richard Feynman in the direct transcripts of his Feynman Lectures on Physics, Volume III, Section 3.4, Identical particles. (Unfortunately, the quote was edited out of the printed version of the same lecture.) "... if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails."
The word "interferes" in this context is a quick way of saying that such objects fall under the rules of quantum mechanics, in which they behave more like waves that interfere than like everyday large objects. In short, when an object becomes so simple that a symmetry assertion of the form F(x) = x becomes an exact statement of experimentally verifiable sameness, x ceases to follow the rules of classical physics and must instead be modeled using the more complex—and often far less intuitive—rules of quantum physics. This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.
Generalizations of symmetry If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. Indeed, A. Connes in his book `Non-commutative geometry' writes that Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the hydrogen spectrum. The notion of groupoid also leads to notions of multiple groupoids, namely sets with many compatible groupoid structures, a structure which trivialises to abelian groups if one restricts to groups. This leads to prospects of `higher order symmetry' which have been a little explored, as follows. The automorphisms of a set, or a set with some structure, form a group, which models a homotopy 1-type. The automorphisms of a group G naturally form a crossed module , and crossed modules give an algebraic model of homotopy 2-types. At the next stage, automorphisms of a crossed module fit into a structure known as a crossed square, and this structure is known to give an algebraic model of homotopy 3-types. It is not known how this procedure of generalising symmetry may be continued, although crossed n-cubes have been defined and used in algebraic topology, and these structures are only slowly being brought into theoretical physics. Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups, yet the different options are indistinguishable during various circumstances
Chapter 2
Group (Mathematics)
The possible manipulations of this Rubik's Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the
group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.
Definition and illustration First example: the integers One of the most familiar groups is the set of integers Z which consists of the numbers ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. 1. For any two integers a and b, the sum a + b is also an integer. In other words, the process of adding integers two at a time always yields an integer, not some other type of number such as a fraction. This property is known as closure under addition. 2. For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.
3. If a is any integer, then 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer. 4. For every integer a, there is an integer b such that a + b = b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following abstract definition is developed.
Definition A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms: Closure For all a, b in G, the result of the operation, a • b, is also in G. Associativity For all a, b and c in G, (a • b) • c = a • (b • c). Identity element There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. The identity element of a group G is often written as 1 or 1G, a notation inherited from the multiplicative identity. Inverse element For each a in G, there exists an element b in G such that a • b = b • a = 1G. The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation a•b=b•a may not always be true. This equation does always hold in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). However, it does not always hold in the symmetry group below. Groups for which the equation a • b = b • a always holds are called abelian (in honor of Niels Abel). Thus, the integer addition group is abelian, but the following symmetry group is not. The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, sometimes a shorthand expression such as "a subset of the group G" is used when what is actually meant is "a subset of the underlying set G of the group (G, •)." Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.
Second example: a symmetry group The symmetries (i.e., rotations and reflections) of a square form a group called a dihedral group, and denoted D4. The following symmetries occur:
id (keeping it as is)
fv (vertical flip)
r1 (rotation by 90° right)
r2 (rotation by 180° right)
r3 (rotation by 270° right)
fh (horizontal flip)
fd (diagonal flip)
fc (counterdiagonal flip)
The elements of the symmetry group of the square (D4). The vertices are colored and numbered only to visualize the operations.
the identity operation leaving everything unchanged, denoted id; rotations of the square by 90° right, 180° right, and 270° right, denoted by r1, r2 and r3, respectively; reflections about the vertical and horizontal middle line (fh and fv), or through the two diagonals (fd and fc).
The defining operation of this group is function composition: The eight symmetries are functions from the square to the square, and two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first a and then b is written symbolically from right to left as b • a ("apply the symmetry b after performing the symmetry a"). The right-to-left notation is the same notation that is used for composition of functions. The group table on the right lists the results of all such compositions possible. For example, rotating by 270° right (r3) and then flipping horizontally (fh) is the same as performing a reflection along the diagonal (fd). Using the above symbols, highlighted in blue in the group table:
fh • r3 = fd. Group table of D4 •
id
r1
r2
r3
fv
fh
fd
fc
id
id
r1
r2
r3
fv
fh
fd
fc
r1
r1
r2
r3
id
fc
fd
fv
fh
r2
r2
r3
id
r1
fh
fv
fc
fd
r3
r3
id
r1
r2
fd
fc
fh
fv
fv
fv
fd
fh
fc
id
r2
r1
r3
fh
fh
fc
fv
fd
r2
id
r3
r1
fd
fd
fh
fc
fv
r3
r1
id
r2
fc
fc
fv
fd
fh
r1
r3
r2
id
The elements id, r1, r2, and r3 form a subgroup, highlighted in red (upper left region). A left and right coset of this subgroup is highlighted in green (in the last row) and yellow (last column), respectively.
Given this set of symmetries and the described operation, the group axioms can be understood as follows: 1. The closure axiom demands that the composition b • a of any two symmetries a and b is also a symmetry. Another example for the group operation is r3 • fh = fc, i.e. rotating 270° right after flipping horizontally equals flipping along the counter-diagonal (fc). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table. 2. The associativity constraint deals with composing more than two symmetries: Starting with three elements a, b and c of D4, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose a and b into a single symmetry, then to compose that symmetry with c. The other way is to first compose b and c, then to compose the resulting symmetry with a. The associativity condition (a • b) • c = a • (b • c) means that these two ways are the same, i.e., a product of many group elements can be simplified in any order. For example, (fd • fv) • r2 = fd • (fv • r2) can be checked using the group table at the right (fd • fv) • r2 = r3 • r2 = r1, which equals fd • (fv • r2) = fd • fh = r1. While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. For instance, subtraction of numbers is not associative: (7 − 3) − 2 = 2 is not the same as 7 − (3 − 2) = 6. 3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form, id • a = a, a • id = a. 4. An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of transformations—identity id, the flips fh, fv, fd, fc and the 180° rotation r2—is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r3 and r1 are each other's inverse, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. In symbols, fh • fh = id,
r3 • r1 = r1 • r3 = id. In contrast to the group of integers above, where the order of the operation is irrelevant, it does matter in D4: fh • r1 = fc but r1 • fh = fd. In other words, D4 is not abelian, which makes the group structure more difficult than the integers introduced first.
History The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884. The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers. The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was pushed by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by pivotal work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians,
classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch crucially impacting many other fields.
Elementary consequences of the group axioms Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity axiom show that the unambiguity of a • b • c = (a • b) • c = a • (b • c) generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. The axioms may be weakened to assert only the existence of a left identity and left inverses. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above.
Uniqueness of identity element and inverses Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element. To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted l and r, in a group (G, •). Then l = l • 1G = l • (a • r) = (l • a) • r = 1G • r =r
as 1G is the identity element because r is an inverse of a, so 1G = a • r by associativity, which allows to rearrange the parentheses since l is an inverse of a, i.e. l • a = 1G for 1G is the identity element
The two extremal terms l and r are equal, since they are connected by a chain of equalities. In other words there is only one inverse element of a. Similarly, to prove that the identity element of a group is unique, assume G is a group with two identity elements 1G and e. Then 1G = 1G • e = e, hence 1G and e are equal.
Division In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation x • a = b. In fact, right multiplication of the
equation by a−1 gives the solution x = x • a • a−1 = b • a−1. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a−1 • b. In general, x and y need not agree. A consequence of this is that multiplying by a group element g is a bijection. Specifically, if g is an element of the group G, there is a bijection from G to itself called left translation by g sending h ∈ G to g • h. Similarly, right translation by g is a bijection from G to itself sending h to h • g. If G is abelian, left and right translation by a group element are the same.
Basic concepts The following sections use mathematical symbols such as X = {x, y, z} to denote a set X containing elements x, y, and z, or alternatively x ∈ X to restate that x is an element of X. The notation f : X → Y means f is a function assigning to every element of X an element of Y. To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed. There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.
Group homomorphisms Group homomorphisms are functions that preserve group structure. A function a: G → H between two groups (G,•) and (H,*) is a homomorphism if the equation a(g • k) = a(g) * a(k) holds for all elements g, k in G. In other words, the result is the same when performing the group operation after or before applying the map a. This requirement ensures that a(1G) = 1H, and also a(g)−1 = a(g−1) for all g in G. Thus a group homomorphism respects all the structure of G provided by the group axioms. Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another (in each of the two possible orders) equal the identity function of G and H, respectively. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element
g of G is equivalent to proving that a(g) • a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.
Subgroups Informally, a subgroup is a group H contained within a bigger one, G. Concretely, the identity element of G is contained in H, and whenever h1 and h2 are in H, then so are h1 • h2 and h1−1, so the elements of H, equipped with the group operation on G restricted to H, indeed form a group. In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole. Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S. In the introductory example above, the subgroup generated by r2 and fv consists of these two elements, the identity element id and fh = fv • r2. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.
Cosets In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a flip is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further flips), i.e. the rotation operations are irrelevant to the question whether a flip has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right coset of H containing g are gH = {g • h, h ∈ H} and Hg = {h • g, h ∈ H}, respectively. The cosets of any subgroup H form a partition of G; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection. The first case g1H = g2H happens precisely when g1−1 • g2 ∈ H, i.e. if the two elements differ by an element of H. Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i.e. for all g in G, gH = Hg, then H is said to be a normal subgroup. One may then simply refer to N as the set of cosets. In D4, the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equal to R, if g is an element of R itself, or otherwise equal to U
= fcR = {fc, fv, fd, fh} (highlighted in green). The subgroup R is also normal, because fcR = U = Rfc and similarly for any element other than fc.
Quotient groups In addition to disregarding the internal structure of a subgroup by considering its cosets, it is desirable to endow this coarser entity with a group law called quotient group or factor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N, the quotient group is defined by G / N = {gN, g ∈ G}, "G modulo N". This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1 = (g−1)N. •
R
U
R
R
U
U
U
R
Group table of the quotient group D4 / R. The elements of the quotient group D4 / R are R itself, which represents the identity, and U = fvR. The group operation on the quotient is shown at the right. For example, U • U = fvR • fvR = (fv • fv)R = R. Both the subgroup R = {id, r1, r2, r3}, as well as the corresponding quotient are abelian, whereas D4 is not abelian. Building bigger groups by smaller ones, such as D4 from its subgroup R and the quotient D4 / R is abstracted by a notion called semidirect product. Quotient and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) flip), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations r 4 = f 2 = (r • f)2 = 1,
the group is completely described. A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups. Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N. Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
Examples and applications
A periodic wallpaper pattern gives rise to a wallpaper group.
The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers. Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered nullhomotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory. In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in
particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.
Numbers Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups. Integers The group of integers Z under addition, denoted (Z, +), has been described above. The integers, with the operation of multiplication instead of addition, (Z, ·) do not form a group. The closure, associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a · b = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of Z has a (multiplicative) inverse. Rationals The desire for the existence of multiplicative inverses suggests considering fractions
Fractions of integers (with b nonzero) are known as rational numbers. The set of all such fractions is commonly denoted Q. There is still a minor obstacle for (Q, ·), the rationals with multiplication, being a group: because the rational number 0 does not have a multiplicative inverse (i.e., there is no x such that x · 0 = 1), (Q, ·) is still not a group. However, the set of all nonzero rational numbers Q \ {0} = {q ∈ Q, q ≠ 0} does form an abelian group under multiplication, denoted (Q \ {0}, ·). Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied. The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and—if division is possible, such as in Q—fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.
Nonzero integers modulo a prime For any prime number p, modular arithmetic furnishes the multiplicative group of integers modulo p. Its elements are integers not divisible by p, considered modulo p, i.e. two numbers are considered equivalent if their difference is divisible by p. For example, if p = 5, there are exactly four group elements 1, 2, 3, 4: multiples of 5 are excluded and 6 and −4 are both equivalent to 1 etc. The group operation is given by multiplication. Therefore, 4 · 4 = 1, because the usual product 16 is equivalent to 1, for 5 divides 16 − 1 = 15, denoted 16 ≡ 1 (mod 5). The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication. The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that a · b ≡ 1 (mod p), i.e. p divides the difference a · b − 1. The inverse b can be found by using Bézout's identity and the fact that the greatest common divisor gcd(a, p) equals 1. In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2 = 6 ≡ 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar to (Q\{0}, ·) above, because it turns out to be the multiplicative group of nonzero elements in the finite field Fp, denoted Fp×. These groups are crucial to public-key cryptography.
Cyclic groups
The 6th complex roots of unity form a cyclic group. z is a primitive element, but z2 is not, because the odd powers of z are not a power of z2. A cyclic group is a group all of whose elements are powers (when the group operation is written additively, the term 'multiple' can be used) of a particular element a. In multiplicative notation, the elements of the group are: ..., a−3, a−2, a−1, a0 = e, a, a2, a3, ..., where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1=(a • a • a)−1 etc Such an element a is called a generator or a primitive element of the group. A typical example for this class of groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1 (and whose operation is multiplication). Any cyclic group with n elements is isomorphic to this group. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1. Some cyclic groups have an infinite number of elements. In these groups, for every nonzero element a, all the powers of a are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to (Z, +), the
group of integers under addition introduced above. As these two prototypes are both abelian, so is any cyclic group. The study of abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.
Symmetry groups Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
Rotations and flips form the symmetry group of a great icosahedron. In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. The JahnTeller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule.
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition. Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.
Hexaaquacopper(II) Ammonia, complex ion, NH3. Its [Cu(OH2)6]2+. symmetry Buckminsterfullerene Cubane C8H8 Compared to a group is of features displays perfectly order 6, octahedral symmetrical shape, icosahedral generated by a symmetry. symmetry. the molecule is 120° rotation vertically dilated by and a about 22% (Jahnreflection. Teller effect).
The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane.
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
General linear group and representation theory
Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the x-coordinate by factor 2. Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics. Representation theory is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of group representations are linear representations, i.e. the group is acting on a vector space, such as the three-dimensional Euclidean space R3. A representation of G on an n-dimensional real vector space is simply a group homomorphism ρ: G → GL(n, R) from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations. Given a group action, this gives further means to study the object being acted on On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.
Galois groups Galois groups have been developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation ax2 + bx + c = 0 are given by
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above. The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
Finite groups A group is called finite if it has a finite number of elements. The number of elements is called the order of the group G. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible swaps of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle. The order of an element a in a group G is the least positive integer n such that a n = e, where a n represents
i.e. application of the operation • to n copies of a. (If • represents multiplication, then an corresponds to the nth power of a.) In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.
More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any finite subgroup H divides the order of G. The Sylow theorems give a partial converse. The dihedral group (discussed above) is a finite group of order 8. The order of r1 is 4, as is the order of the subgroup R it generates. The order of the reflection elements fv etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem. The groups Fp× above have order p − 1.
Classification of finite simple groups Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim quickly leads to difficult and profound mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded to prove the monstrous moonshine conjectures, a surprising and deep relation of the largest finite simple sporadic group—the "monster group"—with certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.
Groups with additional structure Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.
Topological groups
The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom). Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R \ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:
for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above
sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.
Lie groups Lie groups (in honor of Sophus Lie) are groups which also have a manifold structure, i.e. they are spaces looking locally like some Euclidean space of the appropriate dimension. Again, the additional structure, here the manifold structure, has to be compatible, i.e. the maps corresponding to multiplication and the inverse have to be smooth. A standard example is the general linear group introduced above: it is an open subset of the space of all n-by-n matrices, because it is given by the inequality det (A) ≠ 0, where A denotes an n-by-n matrix. Lie groups are of fundamental importance in physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.
Generalizations Group-like structures Totality Associativity Identity Inverses Yes Yes Yes Yes Group Yes Yes Yes No Monoid Yes No No Semigroup Yes Yes No Yes Yes Loop No No No Quasigroup Yes Yes No No No Magma No Yes Yes Yes Groupoid
Category
No
Yes
Yes
No
In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z \ {0}, ·). There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q \ {0}, ·) is derived from (Z \ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.
Chapter 3
Group Action
Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle "acts" on the set of vertices of the triangle by mapping every vertex to another one. In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group
(especially if the set is a vector space and the group acts like linear transformations of the set). A group action is a flexible generalization of the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets, such as the set of vertices, the set of edges and the set of faces of the polyhedron. If G is a group and X is a set then a group action may be defined as a group homomorphism from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that
the permutation of X assigned to the identity element of G is the identity transformation of X; the permutation of X assigned to a product gh of two elements of the group is the composite of the permutations assigned to g and h.
Since each element of G is represented as a permutation, a group action is also known as a permutation representation. The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains widereaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.
Definition If G is a group and X is a set, then a (left) group action of G on X is a binary function
denoted
which satisfies the following two axioms: 1. (gh)·x = g·(h·x) for all g, h in G and x in X; 2. e·x = x for every x in X (where e denotes the identity element of G). The set X is called a (left) G-set. The group G is said to act on X (on the left).
From these two axioms, it follows that for every g in G, the function which maps x in X to g·x is a bijective map from X to X (its inverse being the function which maps x to g-1·x). Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to X. In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms: 1. x·(gh) = (x·g)·h; 2. x·e = x. The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. From a right action a left action can be constructed by composing with the inverse operation on the group. If r is a right action, then
is a left action, since
and
Similarly, any left action can be converted into a right action. Therefore in the sequel we consider only left group actions, since right actions add nothing new.
Examples
The trivial action for any group G is defined by g·x=x for all g in G and all x in X; that is, the whole group G induces the identity permutation on X. Every group G acts on G in two natural but essentially different ways: g·x = gx for all x in G, or g·x = gxg−1 for all x in G. The latter action is often called the conjugation action, and an exponential notation is commonly used for the rightaction variant: xg = g−1xg; it satisfies (xg)h = xgh. The symmetric group Sn and its subgroups act on the set { 1, ... , n } by permuting its elements The symmetry group of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron. The symmetry group of any geometrical object acts on the set of points of that object
The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...). The general linear group GL(n,R), special linear group SL(n,R), orthogonal group O(n,R), and special orthogonal group SO(n,R) are Lie groups which act on Rn. The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group. The additive group of the real numbers (R, +) acts on the phase space of "wellbehaved" systems in classical mechanics (and in more general dynamical systems): if t is in R and x is in the phase space, then x describes a state of the system, and t·x is defined to be the state of the system t seconds later if t is positive or −t seconds ago if t is negative. The additive group of the real numbers (R, +) acts on the set of real functions of a real variable with (g·f)(x) equal to e.g. f(x + g), f(x) + g, f(xeg), f(x)eg, f(x + g)eg, or f(xeg) + g, but not f(xeg + g) The quaternions with modulus 1, as a multiplicative group, act on R3: for any such quaternion , the mapping f(x) = z x z* is a counterclockwise rotation through an angle about an axis v; −z is the same rotation. The isometries of the plane act on the set of 2D images and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g. a function of position with values in a set of colors. More generally, a group of bijections g: V → V acts on the set of functions x: V → W by (gx)(v) = x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it.
Types of actions The action of G on X is called
transitive if X is non-empty and if equivalently 1. for any x, y in X there exists a g in G such that gx = y, 2. Gx = X for all x in X, 3. Gx = X for some x in X. Here, Gx = {g.x | g in G} is the orbit of x under G. sharply transitive if that g is unique; it is equivalent to regularity defined below. n-transitive if X has at least n elements and for any pairwise distinct x1, ..., xn and pairwise distinct y1, ..., yn there is a g in G such that g.xk = yk for 1 ≤ k ≤ n. A 2transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define 2-transitive groups, 3transitive groups, and multiply transitive groups. o
sharply n-transitive if there is exactly one such g. faithful (or effective) if for any two distinct g, h in G there exists an x in X such that g·x ≠ h·x; or equivalently, if for any g≠ e in G there exists an x in X such that g·x ≠ x. Intuitively, different elements of G induce different permutations of X. free (or semiregular) if for all x in X, g.x = h.x only if g = h. Equivalently: if there exists an x in X such that g.x = x (that is, if g has at least one fixed point), then g is the identity. regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g·x = y. In this case, X is known as a principal homogeneous space for G or as a G-torsor. locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if g·x = x for some x and some g in U then g = e. irreducible if X is a nonzero module over a ring R, the action of G is R-linear, and there is no nonzero proper invariant submodule. o
Every free action on a non-empty set is faithful. A group G acts faithfully on X if and only if the homomorphism G → Sym(X) has a trivial kernel. Thus, for a faithful action, G is isomorphic to a permutation group on X; specifically, G is isomorphic to its image in Sym(X). The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G) — a result known as Cayley's theorem. If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g·x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(X). The factor group G/N acts faithfully on X by setting (gN)·x = g·x. The original action of G on X is faithful if and only if N = {e}.
Orbits and stabilizers
In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to which tetrahedron g sends the chosen tetrahedron to. Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx:
The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by
saying x ~ y if and only if there exists a g in G with g·x = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same, i.e. Gx = Gy. The set of all orbits of X under the action of G is written as X /G (or, less frequently: G \X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention. If Y is a subset of X, we write GY for the set { g·y : y ∈ Y and g ∈ G}. We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on Y. The subset Y is called fixed under G if g·y = y for all g in G and all y in Y. Every subset that's fixed under G is also invariant under G, but not vice versa. Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit. For every x in X, we define the stabilizer subgroup of x (also called the isotropy group or little group) as the set of all elements in G that fix x:
This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism G → Sym(X) is given by the intersection of the stabilizers Gx for all x in X. Orbits and stabilizers are closely related. For a fixed x in X, consider the map from G to X given by g ↦ g·x. The image of this map is the orbit of x and the coimage is the set of all left cosets of Gx. The standard quotient theorem of set theory then gives a natural bijection between G /Gx and Gx. Specifically, the bijection is given by hGx ↦ h·x. This result is known as the orbit-stabilizer theorem. If G and X are finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives
This result is especially useful since it can be employed for counting arguments. Note that if two elements x and y belong to the same orbit, then their stabilizer subgroups, Gx and Gy, are conjugate (in particular, they are isomorphic). More precisely: if y = g·x,
then Gy = gGx g−1. Points with conjugate stabilizer subgroups are said to have the same orbit-type. A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
where Xg is the set of points fixed by g. This result is mainly of use when G and X are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element. The set of formal differences of finite G-sets forms a ring called the Burnside ring, where addition corresponds to disjoint union, and multiplication to Cartesian product. A G-invariant element of X is x ∈ X such that g·x = x for all g ∈ G. The set of all such x is denoted XG and called the G-invariants of X. When X is a G-module, XG is the zeroth group cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.
Group actions and groupoids The notion of group action can be put in a broader context by using the associated `action groupoid' associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. Further the stabilisers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. which is a `covering This action groupoid comes with a morphism morphism of groupoids'. This allows a relation between such morphisms and covering maps in topology.
Morphisms and isomorphisms between G-sets If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X → Y such that f(g·x) = g·f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps. If such a function f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case. Some example isomorphisms:
Every regular G action is isomorphic to the action of G on G given by left multiplication. Every free G action is isomorphic to G×S, where S is some set and G acts by left multiplication on the first coordinate. Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G.
With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).
Continuous group actions One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × X → X is continuous with respect to the product topology of G × X. The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions. If G is a discrete group acting on a topological space X, the action is properly discontinuous if for any point x in X there is an open neighborhood U of x in X, such that for which consists of the identity only. If X is a the set of all regular covering space of another topological space Y, then the action of the deck transformation group on X is properly discontinuous as well as being free. Every free, properly discontinuous action of a group G on a path-connected topological space X arises in this manner: the quotient map X ↦ X/G is a regular covering map, and the deck transformation group is the given action of G on X. Furthermore, if X is simply connected, the fundamental group of X / G will be isomorphic to G. These results have been generalised in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of a symmetric square. An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that GA = X. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G. The action of G on X is said to be proper if the mapping G×X → X×X that sends (g,x)↦(gx,x) is a proper map.
Strongly continuous group action and smooth points If is an action of a topological group G on another topological space X, one says that it is strongly continuous if for all , the map g ↦ αg(x) is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous function on X by
.
The subspace of smooth points for the action α is the subspace of X of points x such that g ↦ αg(x) is smooth, i.e. it is continuous and all derivatives are continuous.
Generalizations One can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. Instead of actions on sets, one can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion. One can view a group G as a category with a single object in which every morphism is invertible. A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category. Without using the language of categories, one can extend the notion of a group action on a set X by studying as well its induced action on the power set of X. This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries.
Chapter 4
Regular Polytope
Regular polytope examples
A regular pentagon is a polygon, a twodimensional polytope with 5 edges, represented by Schläfli symbol {5}.
A regular dodecahedron is a polyhedron, a threedimensional polytope, with 12 pentagonal faces, represented by Schläfli symbol {5,3}.
A regular dodecaplex is a polychoron, a fourdimensional polytope, with 120 dodecahedral cells, represented by Schläfli symbol {5,3,3}. (shown here as a Schlegel diagram)
A regular cubic honeycomb is a tessellation, an infinite three-dimensional polytope,represented by Schläfli symbol {4,3,4}.
The 256 vertices and 1024 edges of an 8-cube can be shown in this orthogonal projection (Petrie polygon) In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets [(n − 1)-faces] and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form {a, b, c, ...., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}.
Classification and description Regular polytopes are classified primarily according to their dimensionality. They can be further classified according to symmetry. For example the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and
icosahedron. Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality:
Regular simplex Measure polytope (Hypercube) Cross polytope (Orthoplex)
In two dimensions there are infinitely many regular polygons. In three and four dimensions there are several more regular polyhedra and polychora besides these three. In five dimensions and above, these are the only ones. The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.
Schläfli symbols A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.
A convex regular polygon having n sides is denoted by {n}. So an equilateral triangles is {3}, a square {4}, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value {n/m}, where n and m are co-prime, so a regular pentagram is {5/2}.
A regular polyhedron having faces {n} with p faces joining around a vertex is denoted by {n, p}. The nine regular polyhedra are {3, 3} {3, 4} {4, 3} {3, 5} {5, 3} {3, 5/2} {5/2, 3} {5, 5/2} and {5/2, 5}. {p} is the vertex figure of the polyhedron.
A regular polychoron or polycell having cells {n, p} with q cells joining around an edge is denoted by {n, p, q}. The vertex figure of the polychoron is a {p, q}.
A five-dimensional regular polytope is an {n, p, q, r}. And so on.
Duality of the regular polytopes The dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} and so on.
The vertex figure of a regular polytope is the dual of the dual polytope's facet. For example, the vertex figure of {3, 3, 4} is {3, 4}, the dual of which is {4, 3} — a cell of {4, 3, 3}. The measure and cross polytopes in any dimension are dual to each other. If the Schläfli symbol is palindromic, i.e. reads the same forwards and backwards, then the polyhedron is self-dual. The self-dual regular polytopes are:
All regular polygons, {a}. All regular n-simplexes, {3,3,...,3} The regular 24-cell in 4 dimensions, {3,4,3}. All regular n-dimensional cubic honeycombs, {4,3,...,3,4}. These may be treated as infinite polytopes.
Regular simplices Graphs of the 1-simplex to 4-simplex.
Line segment Triangle
Tetrahedron Pentachoron
Begin with a point A. Mark point B at a distance r from it, and join to form a line segment. Mark point C in a second, orthogonal, dimension at a distance r from both, and join to A and B to form an equilateral triangle. Mark point D in a third, orthogonal, dimension a distance r from all three, and join to form a regular tetrahedron. And so on for higher dimensions. These are the regular simplices or simplexes. Their names are, in order of dimensionality: 0. Point 1. Line segment 2. Equilateral triangle (regular trigon) 3. Regular tetrahedron 4. Regular pentachoron or 4-simplex 5. Regular hexateron or 5-simplex ... An n-simplex has n+1 vertices.
Measure polytopes (hypercubes) Graphs of the 2-cube to 4-cube.
Square
Cube
Tesseract
Begin with a point A. Extend a line to point B at distance r, and join to form a line segment. Extend a second line of length r, orthogonal to AB, from B to C, and likewise from A to D, to form a square ABCD. Extend lines of length r respectively from each corner, orthogonal to both AB and BC (i.e. upwards). Mark new points E,F,G,H to form the cube ABCDEFGH. And so on for higher dimensions. These are the measure polytopes or hypercubes. Their names are, in order of dimensionality: 0. Point 1. Line segment 2. Square (regular tetragon) 3. Cube (regular hexahedron) 4. Tesseract (regular octachoron) or 4-cube 5. Penteract (regular decateron) or 5-cube ... An n-cube has 2n vertices.
Cross polytopes (orthoplexes) Graphs of the 2-orthoplex to 4-orthoplex.
Square
Octahedron 16-cell
Begin with a point O. Extend a line in opposite directions to points A and B a distance r from O and 2r apart. Draw a line COD of length 2r, centred on O and orthogonal to AB. Join the ends to form a square ACBD. Draw a line EOF of the same length and centered on 'O', orthogonal to AB and CD (i.e. upwards and downwards). Join the ends to the square to form a regular octahedron. And so on for higher dimensions. These are the cross polytopes or orthoplexes. Their names are, in order of dimensionality: 0. Point 1. Line segment 2. Square (regular tetragon) 3. Regular octahedron 4. Regular hexadecachoron (16-cell) or 4-orthoplex 5. Regular triacontakaiditeron (Pentacross) or 5-orthoplex ... An n-orthoplex has 2n vertices.
History of discovery Convex polygons and polyhedra The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from ancient Greek mathematicians. The five Platonic solids were known to them. Pythagoras knew of at least three of them and Theaetetus (ca. 417 B.C. – 369 B.C.) described all five. Later, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids. Platonic solids
Tetrahedron Cube
Octahedron Dodecahedron Icosahedron
Star polygons and polyhedra Our understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number. Thomas Bradwardine (Bradwardinus) was the first to record a serious study of star polygons. Various star polyhedra appear in Renaissance art, but it was not until Johannes Kepler studied the small stellated dodecahedron and the great stellated dodecahedron in 1619 that he realised these two were regular. Louis Poinsot discovered the great dodecahedron
and great icosahedron in 1809, and Augustin Cauchy proved the list complete in 1812. These polyhedra are known as collectively as the Kepler-Poinsot polyhedra. Kepler-Poinsot polyhedra
Small stellated Great stellated Great dodecahedron Great icosahedron dodecahedron dodecahedron
Higher-dimensional polytopes
A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. It rotates about the zw plane. It was not until the 19th century that a Swiss mathematician, Ludwig Schläfli, examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in (Schläfli, 1901), six years posthumously, although parts of it were published in 1855 and 1858 (Schläfli, 1855), (Schläfli, 1858). Coxeter (1948) is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six regular convex polytopes in 4 dimensions, five of these correspond to the Platonic solids and the other one is the 24cell. There are exactly three in each higher dimension, which correspond to the tetrahedron, cube and octahedron: these are the regular simplices, measure polytopes and cross polytopes. Descriptions of these may be found in the List of regular polytopes. Also of interest are the nonconvex regular 4-polytopes, partially discovered by Schläfli. By the end of the 19th century, mathematicians such as Arthur Cayley and Ludwig Schläfli had developed the theory of regular polytopes in four and higher dimensions,
such as the tesseract and the 24-cell. The latter are hard to visualise, but still retain the aesthetically pleasing symmetry of their lower dimensional cousins. Harder still to imagine are the more modern abstract regular polytopes such as the 57-cell or the 11-cell. From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives. At the start of the 20th century, the definition of a regular polytope was as follows.
A regular polygon is a polygon whose edges are all equal and whose angles are all equal. A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose vertex figures are all congruent and regular. And so on, a regular n-polytope is an n-dimensional polytope whose (n − 1)dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.
This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.
An n-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to n−1 dimensions, can be mapped to any other such set by a symmetry of the polytope.
So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, or flag, (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:
A regular polytope is one which is transitive on its flags.
In the 20th century, some important developments were made. The symmetry groups of the classical regular polytopes were generalised into what are now called Coxeter groups. Coxeter groups also include the symmetry groups of regular tessellations of space or of the plane. For example, the symmetry group of an infinite chessboard would be the Coxeter group [4,4].
Apeirotopes — infinite polytopes In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges. Nowadays we call them infinite polyhedra or apeirohedra. The regular tilings of the plane {4, 4}, {3, 6} and {6, 3} can also be regarded as infinite polyhedra.
In the 1960s Branko Grünbaum issued a call to the geometric community to consider more abstract types of regular polytopes that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes, that is, regular polytopes with infinitely many faces. A simple example of an apeirogon {∞} would be a zig-zag. It seems to satisfy the definition of a regular polygon — all the edges are the same length, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered.
Regular complex polytopes A complex number has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A complex Hilbert space has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a complex polytope.
Abstract polytopes
The Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face (Grünbaum, 1977). The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.
This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners. The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively-curved (elliptic) hyperspace. A few years after Grünbaum's discovery of the 11-cell, H. S. M. Coxeter independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984). By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called abstract polytopes. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by containment. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes. A geometric polytope is understood to be a realization of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the geometric. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations. The theory has since been further developed, largely by Egon Schulte and Peter McMullen (McMullen, 2002), but other researchers have also made contributions. Regularity of abstract polytopes Regularity has a related, though different meaning for abstract polytopes, since angles and lengths of edges have no meaning. The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes. Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes don't care about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope.
All polygons are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world. Vertex figure of abstract polytopes The concept of vertex figure is also defined differently for an abstract polytope. The vertex figure of a given abstract n-polytope at a given vertex V is the set of all abstract faces which contain V, including V itself. More formally, it is the abstract section Fn / V = {F | V ≤ F ≤ Fn} where Fn is the maximal face, i.e. the notional n-face which contains all other faces. Note that each i-face, i ≥ 0 of the original polytope becomes an (i − 1)-face of the vertex figure. Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures may or may not be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons. The classical vertex figure will, however, be a realisation of the abstract one.
Constructions Polygons The traditional way to construct a regular polygon, or indeed any other figure on the plane, is by compass and straightedge. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21,... Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.
Polyhedra Euclid's Elements gave what amount to ruler-and-compass constructions for the five Platonic solids. (See, for example, Euclid's Elements.) However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.)
Net for icosahedron The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface. If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron. Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available here. Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the Platonic solids (or the Archimedean solids), especially if given a little guidance from a knowledgeable adult. In theory, almost any material may be used to construct regular polyhedra. Instructions for building origami models may be found here, for example. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit.
Higher dimensions
Net for tesseract
A perspective projection (Schlegel diagram) for tesseract
An cut-away cross-section of the 24-cell. In higher dimensions, it becomes harder to say what one means by "constructing" the objects. Clearly, in a 3-dimensional universe, it is impossible to build a physical model of an object having 4 or more dimensions. There are several approaches normally taken to overcome this matter. The first approach is to embed the higher-dimensional objects in three-dimensional space, using methods analogous to the ways in which three-dimensional objects are drawn on the plane. For example, the fold out nets mentioned in the previous section have higherdimensional equivalents. Some of these may be viewed at . One might even imagine building a model of this fold-out net, as one draws a polyhedron's fold-out net on a piece of paper. Sadly, we could never do the necessary folding of the 3-dimensional structure to obtain the 4-dimensional polytope, or polychoron, because of the constraints of the physical universe. Another way to "draw" the higher-dimensional shapes in 3 dimensions is via some kind of projection, for example, the analogue of either orthographic or perspective projection. Coxeter's famous book on polytopes (Coxeter, 1948) has some examples of such orthographic projections. Other examples may be found on the web. Note that immersing even 4-dimensional polychora directly into two dimensions is quite confusing. Easier to understand are 3-d models of the projections. Such models are occasionally found in science museums or mathematics departments of universities (such as that of the Université Libre de Bruxelles). The intersection of a four (or higher) dimensional regular polytope with a threedimensional hyperplane will be a polytope (not necessarily regular). If the hyperplane is moved through the shape, the three-dimensional slices can be combined, animated into a kind of four dimensional object, where the fourth dimension is taken to be time. In this way, we can see (if not fully grasp) the full four-dimensional structure of the fourdimensional regular polytopes, via such cutaway cross sections. This is analogous to the way a CAT scan reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated hologram of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope. Another way a three-dimensional viewer can comprehend the structure of a fourdimensional polychoron is through being "immersed" in the object, perhaps via some form of virtual reality technology. To understand how this might work, imagine what one would see if space were filled with cubes. The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of
himself. If one could travel in these directions, one could explore the array of cubes, and gain an understanding of its geometrical structure. An infinite array of cubes is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (Euclidean) space. However, a 4-dimensional polychoron can be considered a tessellation of a 3dimensional non-Euclidean space, namely, a tessellation of the surface of a fourdimensional sphere (a 4-dimensional spherical tiling).
A regular dodecahedral honeycomb, {5,3,4}, of hyperbolic space projected into 3-space. Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at UIUC has a number of pictures of what one would see if embedded in a tessellation of hyperbolic
space with dodecahedra. Such a tessellation forms an example of an infinite abstract regular polytope. Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its symmetry group is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner.
Regular polytopes in nature Each of the Platonic solids occurs naturally in one form or another: Higher polytopes can obviously not exist in a three-dimensional world. However this might not rule them out altogether. In cosmology and in string theory, physicists commonly model the Universe as having many more dimensions. It is possible that the Universe itself has the form of some higher polytope, regular or otherwise. Astronomers have even searched the sky in the last few years, for tell-tale signs of a few regular candidates, so far without definite results.
Chapter 5
Lie Point Symmetry
Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations. This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically-based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Elie Cartan. Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are translations, rotations and scalings. The Lie symmetry theory is a well-known subject. In it are discussed continuous symmetries opposed to, for example, discrete symmetries. The literature for this theory can be found, among other places, in these notes.
Overview Types of symmetries Lie groups and hence their infinitesimal generators can be naturally "extended" to act on the space of independent variables, state variables (dependent variables) and derivatives of the state variables up to any finite order. There are many other kinds of symmetries. For example, contact transformations let coefficients of the transformations infinitesimal generator depend also on first derivatives of the coordinates. Lie-Bäcklund transformations let them involve derivatives up to an arbitrary order. The possibility of the existence of such symmetries was recognized by Noether. For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by Z.
Applications Lie symmetries were introduced by Lie in order to solve ordinary differential equations. Another application of symmetry methods is to reduce systems of differential equations, finding equivalent systems of differential equations of simpler form. This is called reduction. In the literature, one can find the classical reduction process, and the moving frame-based reduction process. Also symmetry groups can be used for classifying different symmetry classes of solutions.
Geometrical framework Infinitesimal approach Lie's fundamental theorems underline that Lie groups can be characterized by their infinitesimal generators. These mathematical objects form a Lie algebra of infinitesimal generators. Deduced "infinitesimal symmetry conditions" (defining equations of the symmetry group) can be explicitly solved in order to find the closed form of symmetry groups, and thus the associated infinitesimal generators. Let
be the set of coordinates on which a system is defined where n
is the cardinal of Z. An infinitesimal generator δ in the field that has
is a linear operator
in its kernel and that satisfies the Leibniz rule: .
In the canonical basis of elementary derivations
where
is in
for all i in
, it is written as:
.
Lie groups and Lie algebras of infinitesimal generators Lie algebras can be generated by a generating set of infinitesimal generators. To every Lie group, one can associate a Lie algebra. Roughly, a Lie algebra is an algebra constituted by a vector space equipped with Lie bracket as additional operation. The base field of a Lie algebra depends on the concept of invariant. Here only finite-dimensional Lie algebras are considered.
Continuous dynamical systems A dynamical system (or flow) is a one-parameter group action. Let us denote by such a dynamical system, more precisely, a (left-)action of a group (G, + ) on a manifold M:
such that for all point Z in M:
where e is the neutral element of G; in G2,
for all
.
A continuous dynamical system is defined on a group G that can be identified to group elements are continuous.
i.e. the
Invariants An invariant, roughly speaking, is an element that does not change under a transformation.
Definition of Lie point symmetries In this paragraph, we consider precisely expanded Lie point symmetries i.e. we work in an expanded space meaning that the distinction between independent variable, state variables and parameters are avoided as much as possible. A symmetry group of a system is a continuous dynamical system defined on a local Lie group G acting on a manifold M. For the sake of clarity, we restrict ourselves to ndimensional real manifolds where n is the number of system coordinates.
Lie point symmetries of algebraic systems Let us define algebraic systems used in the forthcoming symmetry definition. Algebraic systems Let over the field
be a finite set of rational functions where pi and qi are polynomials in i.e. in variables
with coefficients in . An algebraic system associated to F is defined by the following equalities and inequalities:
An algebraic system defined by
is regular (a.k.a. smooth) if the
system F is of maximal rank k, meaning that the Jacobian matrix at every solution Z of the associated semi-algebraic variety.
is of rank k
Definition of Lie point symmetries The following theorem gives necessary and sufficient conditions so that a local Lie group G is a symmetry group of an algebraic system. Theorem. Let G be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space . Let with define a regular system of algebraic equations:
Then G is a symmetry group of this algebraic system if, and only if,
for every infinitesimal generator δ in the Lie algebra of G. Example Let us consider the algebraic system defined on a space of 6 variables, namely Z = (P,Q,a,b,c,l) with:
The infinitesimal generator
is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely a,b,c and P. One can easily verify that δf1 = f1 − f2 and δf2 = 0. Thus the relations δf1 = δf2 = 0 are satisfied for any Z in that vanishes the algebraic system.
Lie point symmetries of dynamical systems Let us define systems of first-order ODEs used in the forthcoming symmetry definition. Systems of ODEs and associated infinitesimal generators Let
be a derivation w.r.t. the continuous independent variable t. We consider two
sets
and
. The associated coordinate set is
defined by and its cardinal is n = 1 + k + l. With these notations, a system of first-order ODEs is a system where:
and the set specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set X are called state variables, these of Θ parameters. One can associate also a continuous dynamical system to a system of ODEs by resolving its equations. An infinitesimal generator is a derivation that is closely related to systems of ODEs (more precisely to continuous dynamical systems). The infinitesimal generator δ associated to a system of ODEs, described as above, is defined with the same notations as follows:
Definition of Lie point symmetries Here is a geometrical definition of such symmetries. Let be a continuous dynamical its infinitesimal generator. A continuous dynamical system is a Lie point system and symmetry of if, and only if, sends every orbit of to an orbit. Hence, the infinitesimal generator satisfies the following relation based on Lie bracket:
where λ is any constant of linearly independent.
and
i.e.
. These generators are
One does not need the explicit formulas of generators of its symmetries.
in order to compute the infinitesimal
Example Consider Pierre François Verhulst's logistic growth model with linear predation, where the state variable x represents a population. The parameter a is the difference between the growth and predation rate and the parameter b corresponds to the receptive capacity of the environment:
The continuous dynamical system associated to this system of ODEs is:
The independent variable varies continuously; thus the associated group can be identified with . The infinitesimal generator associated to this system of ODEs is:
The following infinitesimal generators belong to the 2-dimensional symmetry group of :
Software There exist many software packages in this area. For example, the package liesymm of Maple provides some Lie symmetry methods for PDEs. It manipulates integration of determining systems and also differential forms. Despite its success on small systems, its integration capabilities for solving determining systems automatically are limited by complexity issues. The DETools package uses the prolongation of vector fields for searching Lie symmetries of ODEs. Finding Lie symmetries for ODEs, the in general case, may be as complicated as solving the original system.
