Two solvability theorems

I. Martin (Irving) Isaacs

Vol. 23 (1967), No. 2, 281–290

DOI: 10.2140/pjm.1967.23.281

Abstract

In this paper we prove two theorems which have certain similarities.

Theorem I. Let G be a group with a cyclic S_p subgroup P such that every p’-subgroup of G is abelian. Then either G has a normal p-complement or else PΔG.

Theorem II. Let G be a group and let p≠2 and q be primes dividing |G|. Suppose for every H < G which is not a q-group or a q’-group that p||H|. If q^a is the q-part of |G| and p > q^a − 1 or if p = q^a − 1 and an S_p of G is abelian then no primes but p and q divide |G|.

Mathematical Subject Classification

Primary: 20.25

Milestones

Received: 8 August 1966

Published: 1 November 1967