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

264

GROUP CHARACTERISTICS

[CH. XIII

COROLLARY 2. Let G be a transitive linear group of order g
in n variables, and let H be a regular substitution group (~ 27)
on g letters simply-isomorphic with G. Then if H be looked
upon as a linear group in g variables, it is intransitive, breaking
up into a number of component groups among which are found
just n which are equivalent to G.
Proof. When a substitution Tt of H other than the identity
is written in matrix form as a linear transformation, every
element in the principal diagonal is zero, since otherwise the
corresponding letter would be replaced by itself in T2. Accordingly, x(Tt)=0 unless Tt is the identity; if T1 is the identity:
(1, 1,..., 1), then x(T1)=g. The transformations of G
being correspondingly SI and S1=(1, 1,..., 1), we have
therefore
g
Lx x(St) * X(TO) =g.
t~x~s,) ~x(r,) = 1
t=l
The corollary now follows by applying Theorem 19.
132. Remark. The propositions of ~ 131 become wider in
scope by an obvious extension of the concept " group." A
group G' of order g' to which another group G of order g = kg'
is multiply isomorphic may be exhibited in such a way as if
it were a group simply isomorphic with G, namely by repeating
each of its transformations h times. For instance, the substitution group of order 6: 1, (ab), (ac), (be), (abc), (acb) is
multiply isomorphic with two of its subgroups: 1; and 1, (ab).
With the concept of " group " extended as indicated above,
we may exhibit the three groups as simply isomorphic in the
following manner:
1 (a, (ac), (bc), (abc), (acb);
1, 1, 1, 1, 1, 1;
1, (ab), (ab), (ab), 1, 1.
EXERCISES
1. Prove that if the regular substitution group EH is broken up into
its ultimate sets of intransitivity with their corresponding component
groups, and if H be multiply isomorphic with a transitive linear group G