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

192 SOLVABLE GROUPS [CH. VIII
EXERCISES
1. A group whose Sylow subgroups of order 2" are cyclic contains
an invariant subgroup of index 2.
Suggestion: Represent the group as a regular group.
2. If a primitive substitution group of degree n is solvable, n must
be a power of a prime and the primitive group must be contained in the
holomorph of the abelian group of order n and of type (1, 1, 1,.. ).
Suggestion: Consider a chief series of composition of the primitive
substitution group.
3. If a transitive group of the prime degree p is solvable, its order is a
divisor of p(p-1).