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MATHEMATICS: L. P. EISENHART Proc. N. A. S. 
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 isomor- 
phisms if it contains a subgroup of degree n and of index n which is non- 
invariant and does not involve any invariant subgroup of the entire group 
besides the identity. From this special but useful theorem it follows di- 
rectly that the symmetric group of degree 6 has outer isomorphisms. 
Various writers established this fact by somewhat laborious special meth- 
ods. 1 It also follows directly from this theorem that the largest imprimi- 
tive groups on six letters which involve 2 and 3 systems of imprimitivity, 
respectively, have isomorphisms which cannot be obtained by trans- 
forming 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 G\ corresponds to a subgroup of degree 
n the conjugates of G\ 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 G\ under G. 
1 Cf. O. Holder, Mathematische Annalen, 46, 1895 (345); W. Burnside, Theory of 
Groups of Finite Order, 1911, p. 209. 
EINSTEIN STATIC FIELDS ADMITTING A GROUP G 2 OF CON- 
TINUOUS TRANSFORMATIONS INTO THEMSELVES 
By L. P. EisEnhart 
Department of Mathematics, Princeton University 
Communicated by O. Veblen, November 3, 1921 
1. For static phenomena in the Einstein theory the linear element of 
the space-time continuum can be taken in the form V 2 dx 0 2 — ds 2 , where 
ds 2 = 2 an, dXi dx k (i t k = 1, 2, 3) (l) 
is the linear element of the physical space 5, and the functions V and a ik 
are independent of x 0 , the coordinate of time. In this paper we determine 
the functions V and a ik satisfying the Einstein equations B ik = Oand 
such that the space 5 admits a continuous group G 2 of transformations 
into itself. Bianchi 1 has shown that any 3-space admitting such a group 
