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

122

PRIME-POWER GROUPS

[CH. V

of order 2B-3 in C such that each operator of Ki is transformed
under K1 into itself multiplied by the various operators of the
corresponding subgroup. Let tl be any non-invariant operator
of K1 and consider all the possible subgroups of order 4 in the
quotient group of K1 with respect to C, such that each of these
subgroups involves the operator corresponding to tl. Any operator (p) of K which is not also in K1 transforms each of these
subgroups into itself multiplied into a subgroup of order 4 contained in C. Let ti,..., t-2 represent a set of operators
of K1 which correspond to a set of independent generating
operators in the given quotient group and assume that
ti-ls2t = Slt2, tl-s3t -s2t3,.. *, t - s i2tl = S-3tsO-.
The subgroup (tl, t2) is transformed by p into itself multiplied
by a group of order 4 which does not involve si. In general,
the subgroup (t, t, a, =2, 3,..., -2, is transformed by
p into itself multiplied by a subgroup of order 4 of C which
does not involve s,_i, and (ti, tta... t,) is transformed
by p into itself multiplied by a subgroup of order 4 which does
not involve s,,_l,,_2.. * * - al, a2. A, a=...,
-2. As p must transform tl into itself multiplied by an operator which is common to all of these subgroups of order 4 and
as si, s2,..., S.-3 are independent generators of a group of
order 2~-32 it results that pt = tp, which is contrary to the hypothesis. That is, we have arrived at a contradiction by assurming that K does not involve an abelian subgroup of order 2~
and hence the theorem under consideration has been proved.
EXERCISES
1. In a group of order pm the order of the commutator subgroup cannot
be greater than pm-2.
2. In a non-abelian group of order p3 each of the non-invariant
operators belongs to a complete system of p conjugates.
3. There is one and only one non-abelian group of order p3, p>2, which
is conformal with the abelian group of type (1, 1, 1).
4. If a non-abelian group of order pm, p>2, contains an operator of
order pm-l its commutator subgroup is of order p and there is only one such
group.