Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

74

ABSTRACT GROUPS

[CH. III

dihedral groups of order 2n and let H1, H2 represent their cyclic
subgroups of order n. Let ti, t2 represent any two elements
of G1, G2 respectively but not contained in H1, H2. The common
order of ti, t2 is 2, and ti, t2 transform corresponding generators
of H1, H2 respectively into corresponding elements, since they
transform all the elements of these subgroups into their inverses.
Hence this theorem includes the known theorem that two
dihedral groups of the same order are always simply isomorphic.
The given illustrative example of the use of the theorem in
question may also serve to point out the way toward a proof.
In fact, if H1, H2 are arranged in a simple isomorphism and
if the products obtained by multiplying corresponding operators by th0, t20 respectively, 3= 1, 2,..., a-1, are placed in
correspondence, we obtain a simple isomorphism between
G1, G2. In fact, th, t2? transform all the corresponding operators of H1, H2 into corresponding operators because they transform generators of HI, H2 in this manner. It may be observed
that this theorem may be employed to prove that two cyclic
groups of the same composite order are simply isomorphic
if it is assumed that two cyclic groups of the same prime order
have this property.
It results immediately from the given theorem that if two
abelian groups of the same order involve only operators of the
same prime order besides the identity, they must be simply
isomorphic. Among the most important simple isomorphisms
are those in which the operators of the same group G are placed
into a (1, 1) correspondence in such a way that an automorphism of G is obtained. We have seen (~ 24) that any operator which transforms G into itself effects an automorphism on
its elements. Moreover, any automorphism of G can always
be brought about by transforming G by some operator which
transforms G into itself. To prove this statement we shall
employ a method which has been illustrated in ~ 14 but which
we desire to exhibit more fully.
Represent G as a regular substitution group and establish
an arbitrary automorphism of G. We may suppose that all
the substitutions begin with the same letter, so that the second