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

136

PRIME-POWER GRO UPS

C H' V

The only groups of order pm, m> 3, whose groups of isomorphisms involve operators of order pm-l are the cyclic group when
p> 2, and the dihedral and the dicyclic groups when p = 2.
Every operator whose order is a power of p in the group of
isomorphisms of G generates with G a group whose order is a
power of p. As G is an invariant subgroup of this group it
results that this group involves a series of invariant subgroups
Go, G1, G2,..., Gm
which are such that t transforms the operators of each of these
subgroups into themselves multiplied by those of a preceding
subgroup. Moreover, whenever t has this property its order is
a power of p. Hence the existence of such a series of subgroups
is a necessary and sufficient condition that an operator t in
the group of isomorphisms of G has for its order a power of p.
In other words,
A necessary and sufficient condition that an operator t in the
group of isomorphisms of a group G of order pm has for its order
a power of p is that t transform every operator in the series of subgroups
Go, G1, G2..., Gm=G
of orders 1, p, p2,..., pm respectively, into itself multiplied by
an operator in the preceding subgroup.
Hence t"' transforms every operator of Ga into itself multiplied by an operator in G _a-i, where -a - 1 is to be replaced
by 0 whenever /3<a+l.
From this theorem it is easy to deduce the following useful
corollary: The order of the commutator of two operators, neither
of which is the identity, of a group G of order pm is less than the
order of the smallest invariant subgroup of G in which either of
these two operators occurs. In particular, if one of these two
operators is of order p and generates an invariant subgroup, it
must be commutative with every operator of G. This special
case may also be regarded as a special case of the theorem that
every invariant subgroup, besides the identity, in a group of
order pm involves invariant operators of order p.