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

~ 107]

IMPRIMITIVE GROUPS

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107. Theorem 9. Let G be an imprimitive linear group in
n variables. These may be chosen in such a manner that they
break up into a certain number of sets of imprimitivity Y1,.,
Yt of m variables each (n=km), permuted according to a transitive substitution group K on k letters, isomorphic with G. That
subgroup of G which corresponds to the subgroup of K leaving one
letter unaltered, say Y1, is primitive as far as the m variables of
the set Y1 are concerned.
If m=l, k=n, then G is said to have the monomial form
or to be a monomial group.
Proof. Let the variables of G break up into say k' sets Y',..., Y,, permuted among themselves according to a substitution group K' on k' letters. This group K' is transitive
(as a substitution group, ~ 12); otherwise G would not be a
transitive linear group. Hence K' contains k'-1 substitutions
S2, S3,..., S, which replace Y1 by Y2, Y3,..., Y
respectively. We shall select k'-1 corresponding transformations of G and denote them by A2, As,..., At. The condition that the determinants of these transformations do not
vanish, implies that the sets contain the same number of
variables n/k'.
There is in K' a subgroup K'1 whose substitutions leave
Y1 unaltered (~ 12). This subgroup, together with the substitutions S2,..., S,, will generate K'. Correspondingly,
G is generated by A2,..., A, and that subgroup G1 of G
corresponding to K'1, and which therefore replaces the variables
of Y1 (say yi, y2,.., ym) by linear functions of the same
variables. If we now fix our attention upon just that portion
of each transformation of G which affects only these m variables
and which plainly forms the transformations of a linear group
[G1] in m variables, we shall prove that if [G1] is not primitive,
then new variables may be introduced into G such that the number
of new sets of imprimitivity is greater than k'.
Accordingly, let the variables of [G1] break up into at least
two subsets of intransitivity or imprimitivity, say Y1),...
Y10'. New variables will now be introduced into the sets Y2,..., Y such that At will replace Y1(l,..., Y1i( by dis