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

~ 70] INVOLVING ONE NON-CYCLIC SYLOW SUBGROUP 181

group of order 4 contained in G, we may construct an ordinary
series of composition of G by using for G2 a subgroup of order
2 contained in G1 but not invariant under G. By omitting G1
from this series there results a series which is not a chief series
of composition of G, since G1 contains the commutator subgroup
of G. The fact that a commutator subgroup series of composition cannot always be obtained by omitting subgroups from
a chief series which is not also a commutator subgroup series
can be illustrated by means of the direct product of the octic
group and a group of order 2.
A solvable group of composite order must be composite, but
not every composite group is solvable. Sometimes the proof
that all the groups which belong to a certain system are composite is equivalent to the proof that they are solvable. This
is clearly the case when the invariant subgroups and the corresponding quotient groups belong to the same system. For
instance, the proof that every group whose order is the product
of distinct prime numbers is composite is equivalent to the
proof that all such groups are solvable. Similarly, the proof
that every group whose order is a power of a prime is composite is equivalent to the proof that all such groups are solvable. On the contrary, the proof that every group whose order
is divisible by 2 but not by 4 is composite does not establish
the fact that such a group is solvable. If it could be proved
that every group of odd order is composite, it would result
from this that every group whose order is not divisible by 4
would be solvable.
70. Groups Involving no More than one Non-cyclic Sylow
Subgroup. One of the most useful theorems as regards solvable groups is the one which affirms that a group is solvable
if it involves either no non-cyclic Sylow subgroup or contains
only cyclic Sylow subgroups besides those whose orders are
divisible by the highest prime which divides the order of the
group. To prove this theorem we assume that the order of such
a group G is written in the form g=pl"lp2a2... pxx, where
pl, P2,..., Px are distinct prime numbers, arranged in
ascending order of magnitude.