Nilpotence of the commutator subgroup in groups admitting fixed point free operator groups

Ernest Edward Shult

Vol. 17 (1966), No. 2, 323–347

DOI: 10.2140/pjm.1966.17.323

Abstract

Let V be a group of operators acting in fixed point free manner on a group G and suppose V has order relatively prime to |G|. Work of several authors has shown that if V is cyclic of prime order or has order four, G′ is nilpotent. In this paper it is proved that G′ is nilpotent if V is non-abelian of order six, but that G′ need not be nilpotent for any further groups other than those just mentioned. A side result is that G has nilpotent length at most 2 when V is non-abelian of order pq, p and q primes (non-Fermat, if |G| is even).

Mathematical Subject Classification

Primary: 20.40

Milestones

Received: 13 July 1964

Published: 1 May 1966