Possible a -Automorphisms of Non-Abelian Groups

VOL,. 15, 1929

MA THEMA TICS: G. A. MILLER

89

POSSIBLE a-AUTOMORPHISMS OF NON-ABELIAN GROUPS BY G. A. MILLZR DUPARTMSNT OF MATHSMATICS, UNIVORSITY OF ILLINOIS Communicated January 12, 1929

It is well known that a necessary and sufficient condition that a group is abelian is that it admits an automorphism in which every operator corresponds to its inverse. Moreover, if every operator of a group can be made to correspond to its second power, or if every such operator can be made to correspond to its third power in an automorphism it can easily be proved that the group is abelian. In the latter and more difficult case a proof is as follows: Let S1, S2 be any two operators of such a group G. It may be assumed that SI,--- S5, S2 -52S2, and that (SlS2)1 = S S5. = = S3S2S, and hence = 52 S2S' Moreover, Hence (5252(S2S1)2 = since is prime to the order is invariant under G 3 That is, S S21 2S21 = SlS2 therefore implies that (S251)2 = S22S0, of G. The relation (SaS2)2 and hence that 552 = S2S. We proceed to prove that it is possible to find an infinite number of non-abelian groups which admit an a-automorphism, differing from the identity automorphism, whenever a has any given integral value except -1, 0, 1, 2 or 3. It is known that when p is any odd prime number then there is one and only one non-abelian group of order pa+l which involves an operator s order pa, a > 1, and that this group involves p operators of order p which transforms every operator of the group into its kpp'- + 1 power, where k has any one of the values 1, 2, .., p-1. The central of this group is of order pa-. When a > 2 and p = 2 there exists also a group of order 2a+1 which satisfies these conditions. The direct product of such a group and any group which involves only operators whose orders divide pa-l will satisfy the same condition. Let a represent any positive integer greater than 3 and suppose that a - 1 = p1 P2 ... pc, X, where pi, P2,. .*, p2 are distinct prime numbers. If X > 1, at least one of these numbers, say p'", is odd and we may extend any group H of order (a - l)p2, which involves the cyclic group of order fco P2a2+1 as a factor of a direct product, by means of an operator t of order P2 which transforms every operator of H into its ath power, and hence every operator of the resulting group into the same power. When X = 1 and Pi = 2, then a, must exceed 1 and hence the desired group can again be constructed. The group obtained by forming the direct product of this group and any group composed of operators whose orders divide a - 1 will obviously have the same property. That is, when a is any positive number greater than 3 it is always possible to construct an infinite number of different non-abelian groups which have the property

An

MA THEMA TICS: G. A. MILLER

PRoc. -N. A. S.

that each of them has an inner non-identity isomorphism in which every operator corresponds to its ath power. To prove that such an infinite system of groups exists also when a is any integer less than -1 we may proceed as follows: Let h = 2 - a and assume that h -1 = p1l p2 2,. ., pe", where the symbols pi, p2, * * *, pX satisfy the same condition as in the preceding paragraph. The group H is constructed just as before but the operator t transforms every operator of H into its (p2 - 1) (h - 1) + 1 power. If we subtract the order of H from the index of this power there results a. That is, this H has an inner automorphism in which every operator corresponds to its ath power, where a is any negative integer less than -1. This completes a proof of the following theorem: If a is any given integer with the exception of the five integers, -1, 0, 1, 2, 3, then it is possible to construct an infinite number of distinct non-abelian groups which have the property that every operator corresponds to its ath power in a non-identity inner isomorphism of each one of these groups. When every operator of a group G can be made to correspond to its ath power in an automorphism of G then the (a - 1)th power of every operator of G is found in its central, and every operator of G can be made to correspond to its (a - 1)th power in an isomorphism of G'. When G is non-abelian this isomorphism cannot be an automorphism of G since the group formed by these a - 1 powers is abelian. Hence it results that when a non-abelian group admits an a-automorphism it must involve an operator besides the identity whose order divides a - 1, and the operators whose orders divide a - 1 must constitute an invariant subgroup of the group, which gives rise to an abelian quotient-group. While this subgroup is composed of the operators of G which remain invariant under the a-automorphism in question it is not necessarily composed of invariant operators of G. Moreover, it does not necessarily contain all the invariant operators of G. As an instance of a group which involves invariant operators which are not invariant under a possible a-automorphism we may cite the nonabelian group of order 64 which contains two cyclic subgroups of order 32. It is obvious that this group admits an a-automorphism in which a = 5 and that the operators of order 8 which appear in its cyclic subgroups of order 16 are in its central, but they are not invariant under this a-automorphism. It should be noted that this is an outer a-automorphism. In every inner a-automorphism all the invariant operators of the group obviously correspond to themselves under this automorphism and hence the order of every such operator is a divisor of a - 1. It was proved above that when a does not have one of the five values, -1, 0, 1,. 2, 3, then it is always possible to find an infinite number of groups which admit a non-identity inner a-automorphism. It is now easy

VOL,. 15, 1929

MA THEMA TICS: W. L. A YRES

91

to prove that whenever a does not have one of these five values then it is also always possible to find an infinite number of groups which admit an outer a-automorphism. In fact, whenever a > 3 it is obviously possible to construct a non-abelian group which involves only operators whose orders divide a - 1. By forming the direct product of this group and any group of prime order greater than a there results a group which admits outer a-automorphisms. Whenever a < -1 we can construct an infinite number of non-abelian groups composed of operators whose orders divide 2 - a. The direct product of such a group and a group of prime order greater than 2 - a evidently admits outer a-automorphisms. That is whenever a does not have one of the five values, -1, 0, 1, 2, 3, it is possible to find an infinite number of non-abelian groups for any given value, of a such that each of them admits an outer a-automorphism. It may be added that whenever a group admits an a-automorphism, where a - 1 is not divisible by more than two distinct prime numbers, it must be solvable in view of the fact that if the order of a group is not divisible by more than two distinct prime numbers it must be solvable. 1 J. W. Young, Trans. Amer. Math. Soc., 3, 186 (1902).

ON CONTINUOUS CURVES HAVING CERTAIN PROPERTIES' By W. L. Ayv"S2 DUPARTMB3NT OF PURE MATHZIMATICS, UNIVERSITY or TExAs

Communicated January 5, 1929

A continuous curve is a connected im kleinen continuum (bounded or not). A continuous curve M is cyclicly connected if every two points of M lie together on some simple closed curve of M.3 A subset C of M is a maximal cyclic curve of M if C is a cyclicly connected continuous curve and M contains no cyclicly connected continuous curve of which C is a proper subset.3 In this note we shall consider continuous curves which have the property that every arc of the continuous curve is a subset of a simple closed curve of the continuous curve. Secondly, we shall consider those in which every arc is a subset of an open curve; and finally, those in which every arc is a subset of a ray. We shall find that these restrictions give much simpler types of continuous curves than might be expected. THHOREM 1. In order that every arc of a plane continuous curve M be a subset of a simple closed curve of M it is necessary and sufficient that M be either a simple closed curve or the entire plane. Proof.-It is evident that the condition is sufficient. We shall show that