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

~~ 114-1151 PRIMITIVE COMPOSITE GROUPS

237

we operate successively by the transformations of H and by
T. Examining the various possibilities we find that (1) could
not be distinct from X1X2X3 =0 unless H is the particular group
generated by the transformations
St=(1, W, W2), S2 = (,, C) (W)=1).
There are then four invariant triangles for (C), namely:
~(2). ~xlx2x3 = 0;
(X1 +X2 + ox3) (xl X2 + +c20x3) (x1 + +02x2+ OX3) =0
(0=1, c or a2).
The same triangles are invariants of (D) if this is generated
by (C) in the form just given and the following special form of R:
X1= - 1, X2 -X 3, X3 -X 2.
114. Groups Having Invariant Intransitive Subgroups. All
such groups are intransitive or imprimitive. This follows from
the fact that the type (B) has a single linear invariant xl,
which is therefore also an invariant of a group containing (B)
invariantly; * and the fact that a group containing (A) invariantly cannot be primitive by the lemma, ~ 108.
115. Primitive Groups Having Invariant Imprimitive Subgroups. It was shown above that the types (C) and (D)
possess either one or a set of four invariant triangles. If they
possess only one such triangle, a group containing one of these
types invariantly would of necessity also leave invariant that
triangle, as may be easily proved. That such a group may
be primitive, it is therefore necessary that (C) and (D) possess
the four invariants (2).
Let us therefore assume a group G permuting among themselves the triangles (2), which we shall for brevity denote
respectively by ht tt2, t3, t4 in the order as they are listed in (2).
* Let V be any transformations of a group containing (B) invariantly, and
T any transformation of (B). Then VTV- = T1 belongs to (B), and if we put
(xi) T=LaXl, (xl) V= y, we have
(y)T= (y)V- T1,V= ay,
so that y is an invariant of T. But, xi being the only linear invariant for (B),
it follows that y= cxi.