Voi, 7, 1921 
MATHEMATICS: G. A. MILLER 
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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 Si, Si are any two corresponding substitutions 
such that in Si the letter e is followed by the letter /, and if 5 2 , S 2 ' are two 
other corresponding substitutions in which the letter a is followed by e 
then Si is obtained by multiplying the inverse of S 2 into one of the g/n 
substitutions in which a is followed by /. Hence it results that in S/ 
the letter e is also followed by the /. That is, Si and Si are identical. 
In this proof it was assumed that e is not a and that / is not a. If e were 
a then a would be replaced by / in Si by hypothesis. If / were a then in 
the inverse of Si the letter a would be replaced by e and hence in the in- 
verse of 5/ 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 trans- 
forms G into itself. 
If the subgroup Gi had corresponded in the given automorphism to 
a conjugate of G\ but different from Gi there would be a substitution in 
G which would transform Gi into this conjugate. The inverse of this 
substitution into t would transform G into itself and effect the auto- 
morphism in question. It has therefore been proved that every automor- 
phism 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 sub- 
stitutions of the largest group on the letters of G which contains G invari- 
antly. As a particular case of this theorem we have the well known re- 
sult that when G is regular it is transformed into all its possible auto- 
morphisms under its holomorph. 
From the theorem proved in the preceding paragraph it is easy to de- 
duce a method for finding the order of the group of isomorphisms of the 
transitive substitution group G. If Gi is of degree n — a the largest group 
on the letters of G which contains G invariantly involves exactly a sub- 
stitutions 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 
Gi in some possible automorphism of G. 
A necessary and sufficient condition that G contains more than one 
