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

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SERIES OF COMPOSITION

179

following it is an abelian group which involves only operators
of prime order besides the identity.
This abelian group is therefore of type (1, 1, 1,. ).
If the order of this quotient group is pa, we must evidently
insert a conjugates of H1 in order to obtain an ordinary series
of composition from the given chief series. The first of these
can be chosen in (pa"- )/(p-1) different ways.
When Go is insolvable, the given method of proof leads directly to the results that the quotient group of any one of the
groups, in the given chief series, with respect to the one immediately following it, is a direct product of simple groups of composite order which are simply isomorphic. If these simply
isomorphic simple groups are of prime order, we have the result
expressed in the preceding theorem. It is easy to prove that
the totality of the quotient groups of each group of a chief
series of composition with respect to the one following it is an
invariant of the group. In fact this proof is practically the
same as the proof of the fact that the factors of composition
of any group is an invariant of the group.
The theorem that the quotient group of any group in a chief
series of composition, with respect to the one which follows it,
is a power of a simple group, results also directly from the fact
that this quotient group cannot involve a characteristic subgroup. The given method of proof leads directly to the theorem
that a necessary and sufficient condition that a group does not
contain a characteristic subgroup is that this group is a power
of a simple group. If this theorem had been assumed as known,
the fact that each of the given quotient groups is a power of a
simple group would not have required any proof.
If we form the successive commutator subgroups of a solvable group Go we obtain a third series
(C) Go, G"1, G"2,..., G",=l.
As the quotient group of each of these groups with respect to
the one which immediately follows it is abelian and as each one
of the successive commutator subgroups is invariant under
Go, it results that as regards the three series A, B, C we have