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

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~ 69] SERIES OF COMPOSITION 177
a necessary and sufficient condition that Go be solvable is that
for a finite value of X, Gx= 1. It is evident that this implies
that Gx_- is abelian and that the order of G, is less than that
of Ga-i whenever a X. The given condition for the solvability of Go is therefore equivalent to saying that a necessary
and sufficient condition that a group be solvable is that none
of its successive commutator subgroups besides the identity
is a perfect group (cf. ~ 29).
That this second definition of a solvable group is equivalent
to the first, follows immediately from the fact that if a group
has an invariant subgroup of prime index, this subgroup must
include the commutator subgroup of the group, and if the
order of the commutator subgroup of a group is less than the
order of the group, there must be an invariant subgroup of prime
index in the group, since the commutator quotient group is
always abelian.
While every simple group of 'composite order is evidently
a perfect group, there are perfect groups which are composite.
EXERCISES
1. The smallest perfect group which is not also simple is of order 120.
2. The factors of composition of the symmetric group of degree n,
nO4, are 2 and n!/2.
3. Every perfect group besides the identity is insolvable, but an insolvable group is not necessarily perfect.
4. Every subgroup of a solvable group is solvable.
5. Each one of the series of successive commutator subgroups is invariant under the original group.
6. Every solvable group of composite order contains an invariant
subgroup which is abelian and whose order exceeds unity.
69. Series of Composition. If each one of the series of groups
(A) Go, G1, G2,..., Gp=l,
excluding the first, is a maximal invariant subgroup of the one
which immediately precedes it, the series is said to be an
ordinary series of composition of Go. For brevity an ordinary
series of composition is often called a series of composition.
A necessary and sufficient condition that Go be a simple group