itself but are not found in G. It will be proved that these automorphisms together with the inner ones constitute all the possible automorphisms of G in which the subgroups composed separately of all the substitutions of G omitting a particular letter correspond to such subgroups. The third category of automorphisms of G is composed of all those in which the subgroups composed of all the substitutions of G which omit a given letter correspond to subgroups of degree n. Such automorphisms are not always possible. In fact it may happen that I is simply isomorphic with the group of inner isomorphisms of G. In this case only the first category of automorphisms actually exists. A necessary and sufficient condition that the first category of automorphisms reduces to the identity is that G is abelian. To prove the theorem noted at the end of the first paragraph we may consider any possible automorphism of G in which the subgroup G1 composed of all the substitutions of G which omit a given letter a corresponds to itself while some of the substitutions of this subgroup do not necessarily correspond to themselves. In this automorphism every substitution of G which involves the letter a must correspond to such a substitution, and we shall suppose that the substitutions are represented in such a way that this letter appears first. There are g/n, g being the order of G, substitutions of G in which the letter a is followed by any other letter b. If in one of the corresponding substitutions under the said automorphisms a is followed by d then this must be the case in each of the g/n substitutions which correspond to the substitutions beginning with the letters ab. If this were not the case as regards some two such corresponding substitutions then the inverses of the former of the two given corresponding substitutions into the latter would give two corresponding substitutions of which the former would omit the letter b while the latter would involve the letter d. This is impossible since the subgroup composed of all the substitutions which omit b must correspond to the subgroup composed of all the substitutions which omit d in the automorphism under consideration as a result of the fact that a substitution beginning with ab corresponds to one beginning with ad. From the preceding paragraph it results that the second letters in all of the corresponding substitutions involving the letter a in the said automorphism determine a single substitution t on the letters of G. This substitution does not involve the letter a. It transforms G into a simply isomorphic group G' such that in the corresponding substitutions involving a the second letters are exactly the same as they are in the second group of the automorphism under consideration. If G were not identical with this second group, as a substitution group, it would be possible to establish a simple isomorphism between G and another substitution group G' such that every substitution involving a would correspond to such a substitution and that the second letter in all these corresponding substitutions would be the same in every case. In this simple isomorphism every two corresponding substitutions would be identical. For, if S1, Si' are any two corresponding substitutions such that in S1 the letter e is followed by the letter ƒ, and if S2, S2' are two other corresponding substitutions in which the letter a is followed by e then S, is obtained by multiplying the inverse of S2 into one of the g/n substitutions in which a is followed by f. Hence it results that in S1' the letter e is also followed by the f. That is, S1 and Si' are identical. In this proof it was assumed that e is not a and that ƒ is not a. If e were a then a would be replaced by ƒ in S1' by hypothesis. If ƒ were a then in the inverse of S1 the letter a would be replaced by e and hence in the inverse of S1' the letter a would be replaced by e. Therefore the theorem again requires no further proof. Hence it has been established that G and G' are identical, and that the automorphism under consideration can be effected by transformed G by the substitution t which therefore transforms G into itself. If the subgroup G1 had corresponded in the given automorphism to a conjugate of G1 but different from G1 there would be a substitution in G which would transform G1 into this conjugate. The inverse of this substitution into t would transform G into itself and effect the automorphism in question. It has therefore been proved that every automorphism of G in which the subgroups composed of all the substitutions of G which omit a given letter correspond to such subgroups is effected by substitutions of the largest group on the letters of G which contains G invariantly. As a particular case of this theorem we have the well known result that when G is regular it is transformed into all its possible automorphisms under its holomorph. From the theorem proved in the preceding paragraph it is easy to deduce a method for finding the order of the group of isomorphisms of the transitive substitution group G. If G1 is of degree na the largest group on the letters of G which contains G invariantly involves exactly a substitutions which are commutative with every substitution of G and hence this largest group transforms the substitutions of G according to a group whose order is the order of this largest group divided by a. This group of transformation is simply isomorphic with the quotient group of this largest group with respect to its invariant subgroup composed of the a substitutions which are commutative with every substitution of G. The order of the I of G is therefore equal to the order of this quotient group multiplied by the number of the different sets of conjugate subgroups of G which are such that each set involves a subgroup which corresponds to G1 in some possible automorphism of G. A necessary and sufficient condition that G contains more than one such set of conjugate subgroups is that it contains subgroups of degree n and of index n with respect to which it can be represented as a transitive substitution group which is identical with G. In particular, if a given abstract group is simply isomorphic with only one group in a complete list of transitive groups of degree n then this transitive group has outer isomorphisms if it contains a subgroup of degree n and of index n which is noninvariant and does not involve any invariant subgroup of the entire group besides the identity. From this special but useful theorem it follows directly that the symmetric group of degree 6 has outer isomorphisms. Various writers established this fact by somewhat laborious special methods. It also follows directly from this theorem that the largest imprimitive groups on six letters which involve 2 and 3 systems of imprimitivity, respectively, have isomorphisms which cannot be obtained by transforming these groups by substitutions on their own letters. While the group of inner isomorphisms is an invariant subgroup of I whenever it does not coincide with I it should not be inferred that the subgroup of I which corresponds to all the automorphisms of G which can be obtained by transforming G by substitutions on its own letters is always an invariant subgroup of I. In fact, this is not the case when G is the generalized dihedral group of order 16 involving the abelian group of type (2, 1) represented as a transitive group of degree 8. When G admits automorphisms in which G1 corresponds to a subgroup of degree n the conjugates of G1 are transformed under I according to an imprimitive substitution group. One of the systems of imprimitivity of this group is composed of the conjugates of G1 under G. 1 Cf. O. Hölder, Mathematische Annalen, 46, 1895 (345); W. Burnside, Theory of Groups of Finite Order, 1911, p. 209. EINSTEIN STATIC FIELDS ADMITTING A GROUP G2 OF CONTINUOUS TRANSFORMATIONS INTO THEMSELVES BY L. P. EISENHART DEPARTMENT of MathemaTICS, PRINCETON UNIVERSITY 1. For static phenomena in the Einstein theory the linear element of the space-time continuum can be taken in the form V2dx2-ds2, where ds2 = 2 aik dx; dxk (i, k = 1, 2, 3) (1) O is the linear element of the physical space S, and the functions V and aik are independent of xo, the coördinate of time. In this paper we determine the functions V and aik satisfying the Einstein equations Bik = 0 and such that the space S admits a continuous group G2 of transformations into itself. Bianchi1 has shown that any 3-space admitting such a group is of one of two types: (I) ds2 = dx12 + adx22 + 2ẞdx2dx3 + ydx32, a, B, y being functions of x1 alone, the operators of the group being X1 =d/dx2, X2 = d/dx3; (II) ds2 = dx12 + adx22 + 2(ẞ — ax2) dx2dx3 + (αx22 - 2ẞx2 + y)dx32, a, B, y being functions of x alone, the operators of the group being X1 = d/dxз, X2 2. By definition 3 1 3 = et3d/dx2. where the Christoffel quantities are calculated with respect to the linear element of the space-time continuum. When this linear element is taken in the form V2dxo2-ds2 as in § 1, the equations Bik O reduce to the six equations2 = Lik Σaik Bik = 0, i, k where a is the cofactor of air in the determinant of the quantities aik divided by this determinant; the Christoffel symbols are calculated with respect to (1), and By assuming that S is referred to a triply-orthogonal system, so that we find that when B1 = 0, then all the functions (il, hk) are equal to zero. ik This being the condition that the space S is euclidean, we have the theorem: A necessary and sufficient condition that a 3-space be euclidean is that the functions Bik = 0. 3. If we put δ αγ γ α'β d = αy — ß2, A = ß'y — By', B = y'a — ya', C = a'ß — aß', (where primes indicate differentiation with respect to x1) we find that all of the Christoffel quantities for the form (I) vanish except the following: The problem reduces to the integration of (3), when a, B, y are subject to the condition (6). We separate the discussion into the four cases, when V is independent of x2 and x3; of either x2 or x3; of x1; involves all three variables. Expressing the conditions of integrability of (3) we are led to linear partial differential equations of the first order and these are ultimately solved. The first and second cases are the only ones giving solutions; in the course of the investigation it is shown that in both cases a change of variables can be made so that ß = 0. The solutions of the first type are: where k is any constant. Solutions of this kind have been found by Kasner. For the second case the linear element of S may be written where a and b are constants. This is the solution found by Levi-Civita3 |