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

134

PRIME-POWER GROUPS

[CH. V

4. There are just three groups of order 34 each of whichcontains
only three cyclic subgroups of order 9.
53. Some Properties of the Group of Isomorphisms of a Group
of Order pm. If Ga is any invariant subgroup of G, then Ga
contains at least one invariant subgroup of G whose order is
an arbitrary divisor of the order of Ga. Let
Go, G1, 2,..., Gm
be a series of invariant subgroups of G whose orders are respectively 1, p, p2,..., p", so that Gm —G; and suppose that each
of these subgroups except the last is included in the one which
follows it. Let t be any operator whose order is a power of p
in the group of isomorphisms of G. Such an operator is said
to effect a p-isomorphism of G. Since t and G generate a group
whose order is a power of p, we may suppose that t transforms
all of the operators of each one of the given series of subgroups
into themselves multiplied by those of the preceding subgroup.
That is, the commutator of t and any operator of Go is in GO-1,
3=1, 2,...,m.
We proceed to prove that the order of t cannot exceed pm-l.
In fact, if tA is any operator of G0 in the above series, it results
from the given conditions that
t- ltt=to_t-a, t- pttp =t_,t.-2
Hence tp"m must be commutative with every operator of G,
and, as t is an operator in the group of isomorphisms of G, it
results from this that the order of t is a divisor of pm-'. In
other words, the pm-l power of every operator whose order is a
power of p in the group of isomorphisms of a group of order pm
is the identity. When G is the cyclic group of order pm, p> 2,
it is known that the group of isomorphisms of G involves operators of order pm-l, viz., those operators which give rise to
commutators of order pm-l.
We proceed to prove that the group of isomorphisms of a
non-cyclic group of order pm, p>2 and m>3, cannot involve
any operator of order pm-l. If G is such a group it follows from
the preceding section that we may assume that G2, in the above