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

~ 94 REDUCIBLE AND IRREDUCIBLE GROUPS

211

that is, if a certain number of the n variables, say xi,..., x,
where m<n, are transformed into linear functions of themselves by every transformation of G.
For instance, a group in two variables is reducible if (either directly
or after a proper change of variables) all the matrices are of type
(b c]
If this is not the case, the group is said to be irreducible.
We shall say that the m variables xl,..., Xm form a reduced
set for G.
THEOREM 6. A reducible group G is intransitive, and a
reduced set constitutes one of the sets of intransitivity of G.
Applied to the illustration above, the theorem asserts that the group
there given can be written in the form
(a 0
0 cJ
by a suitable choice of variables.
Proof. The group G has an Hermitian invariant, which
by the change of variables specified in ~ 91 may be written:
ylyl+y2y2+ ~ ~ ~ +ynyn.
Making the corresponding changes in G, this group is still
seen to be of the form
G' 0
G" G"'
namely,
ys=asly'l+.~ +asy' (s=1, 2,., m),
y,=aly'l+... +aty'm+... +atny'n
(t=m+l, m+2,..., n).
Applying the conditions (8) and (9), ~ 93, and writing Cst for
the product astast, we obtain, among others, the following 2(n-m)
equations:
Cm+lv+. ~. +cn =1 (v=m+l, m+2,... n),
cwi +... +cw=l (w=m+l, m+2,.., n).