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

~ 551 CONSTRUCTION OF GROUPS OF ORDER pm ~ 139

possible groups of order pm by determining all the regular
substitution groups of this order. We proceed to indicate a
method for constructing all the possible regular groups of
order pm on condition that all those of order pm-l are known.
Since every regular group of order pm contains an invariant
intransitive subgroup of order pm"- which is formed by a simple
isomorphism between p regular groups of this order, we may
first determine all the possible regular groups of order pm which
involve a given group H of order pm-.
Let the p transitive constituents of H be H1, H2,..., Hp.
A group G of order pm which contains H is generated by H and
some substitution ti which permutes the p systems of intransitivity of H cyclically, has its pth power in H, and transforms
H into itself. Let t be a substitution of order p which permutes
the systems of H transitively and is commutative with every
substitution of H. As it may be assumed that ti and t permute
these systems in the same way, we conclude that tit-~ is a substitution s which does not permute any of the p transitive constituents of H. That is, ti = st, where it may be assumed that t
transforms the corresponding letters of each of the p systems
among themselves. As t can readily be obtained from the p
constituents belonging to H1, H2,..., Hp of any substitution in H it remains only to determine s = iS2... Sp, where
Si, S2,..., Sp are the constituents of s belonging to H1, H2,.., Hp respectively. As si, s2,..., sp transform each
of the constituents H1, H2,..., Hp into itself they must be
found in the holomorphs of H1, H2,..., Hp, respectively.
We proceed to prove that S2, S3,..., Sp may be assumed
to be corresponding substitutions in the groups of isomorphisms
of H2, H3,.., H, respectively, while sl is one of the pm-l
substitutions obtained by multiplying the substitution in the
group of isomorphisms of Hi which corresponds to S2, S3,...
sp by the pA-1 substitutions which are commutative with
H1 and involve only letters of Hi. This theorem follows
-almost directly from the fact that the conjoint of a regular
group has the same group of isomorphisms as the regular group
itself, since both are invariant subgroups of the holomorph of