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

230

SOME SPECIAL TYPES OF GROUPS

[CH. XI

tinct subsets Yl,..., Y(0. In this manner the variables
of G will be divided into lk' subsets, and it remains for us to
prove that any transformation S of G will permute these subsets among themselves; that is, S will transform the variables
from any one subset into linear functions of the variables
of one of these subsets.
Let S replace Y, by YV. Then ASA-1I=T transforms
Y1 into itself; that is, T is a transformation of G1 and will
therefore permute among themselves the subsets Yl),...,
Y1(l). It follows that the transformation A,,-TA =S will
transform any subset of Ya into some subset of YV, and the
proposition is proved.
We can therefore keep on changing the variables so as
to increase the number of sets of imprimitivity, until the sets
contain just one variable each, or until the group [G1] is primitive. The theorem is therefore proved.
108. Lemma. A linear group G having an invariant abelian
subgroup H whose transformations are not all similarity-transformations is either intransitive or imprimitive.
Proof. Write H in canonical form. The variables can then
be arranged into sets having the property that a transformation of H affects all the variables of any one set by the same
constant factor.
To illustrate, let H be generated by the transformations
T1= (al, oil, ai1, 01, a02) (C1i;a2),
T2=(01, 13, 12, 32,,2) (132)2).
Here we have three sets: X= (xl, x2), Y= (X3, X4), Z= (X5).
Then it is readily proved that G permutes these sets among
themselves. Thus, in the illustration given, let S be a transformation of G and T,, Tj of H, and let S-1TS=T. Now
suppose that the variables of X are transformed by S into
two variables yi, y2 forming a set X'; we must then prove that
X' is either X or Y. We have
(X1, X2)TtS=(X1, x2)STJ,
or
Ca(yi, y2)=(yi, y2)T,.