326 
MATHEMATICS: G. A. MILLER 
Proc. N. A. S. 
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 G\ com- 
posed 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 im- 
possible 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 auto- 
morphism 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 in- 
volving 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 f such that every substitution involving a would correspond to such a 
