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

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THEIR GROUP OF ISOMORPHISMS

135

series of invariant subgroups, is non-cyclic. If t is an operator
in the group of isomorphisms of G and if the order of t is a power
of p, it results that tp is commutative with every operator of
G2 and that it transforms the operators of G into themselves
multiplied by operators in Gm-2. Similarly tp2 is commutative
with every operator in G3 and transforms every operator in Gm
into itself multiplied by an operator in Gm —3. In general, tpa
transforms every operator in G into itself multiplied by operators
in G_-a-1. As tp transforms the operators of G into itself
multiplied by operators in G2 and as these operators are commutative under this power of t, it results that
m-2
t2' =1.
That is, the order of every operator in a Sylow subgroup of order
pm of the group of isomorphisms of a group of order pm is a divisor
of pm-2 whenever p> 2 and m> 3.
The last theorem evidently also applies to all groups of order
2m which involve a non-cyclic invariant subgroup of order 4,
and it is well known to be true as regards the cyclic group of
order 2". It is, however, not true as regards the dihedral or
the dicyclic group of order 2m, since there is evidently an operator of order 2"'-1 in the group of isomorphisms of each of these
groups; viz., the operator which is commutative with each
operator of the cyclic subgroup of order 2m-1 but transforms
the non-invariant operators of orders 2 and 4 respectively into
themselves multiplied by an operator of order 2m'-1. These
two infinite systems of groups therefore are instances of groups
whose groups of isomorphisms contain operators of the largest
possible order in accord with the theorem of the second paragraph of the present section.
As the remaining group of order 2m which does not involve
an invariant non-cyclic subgroup of order 4 involves 2m-3 noninvariant cyclic subgroups of order 4, its group of isomorphisms
may be represented as an intransitive substitution group on
2m-2 letters. From this it follows directly that its group of
isomorphisms cannot involve any operator of order 2m-1,
m>3. Hence we have proved the theorem: