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

CHAPTER VIII

SOLVABLE GROUPS *
68. Introduction. A group is said to be solvable if, and
only if, it contains a series of invariant subgroups such that the
last of the series is the identity and the index of each of these
subgroups under the next larger subgroup is a prime number.
For instance, the symmetric group of order 24 contains an
invariant subgroup of index 2, this subgroup contains an invariant subgroup of index 3, this subgroup is the four-group and
contains an invariant subgroup of index 2, and the identity is
of index 2 under this last subgroup. Hence the symmetric
group of order 24 is solvable. The numbers 2, 3, 2, 2 are said
to be its factors of composition. In general, the factors of
composition of a group are the indices of the successive largest
invariant subgroups. For example, the symmetric group of
order 120 contains an invariant subgroup of index 2, but this
subgroup involves no invariant subgroup besides the identity.
Hence the symmetric group of order 120 is insolvable and has
2 and 60 for its factors of composition. Every abelian group is
evidently solvable.
The terms solvable and insolvable as applied to groups of
finite order are transferred from the theory of equations. An
algebraic equation is solvable by rational processes in addition
to root extractions whenever the group of the equation is solvable and only then (cf. Part III). It should be observed that
an invariant subgroup of an invariant subgroup is not necessarily an invariant subgroup of the entire group. For instance,
the invariant subgroup of order 2 used in connection with the
* In H. Weber's Lehrbuch der Algebra, solvable groups are called metacyclic.
In the present work we use the term metacyclic with its older meaning to represent the holomorph of the group of order p. See p. 12.
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