Vol. 17, No. 2, 1966

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

Ernest Edward Shult

Vol. 17 (1966), No. 2, 323–347

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, Gis nilpotent. In this paper it is proved that Gis nilpotent if V is non-abelian of order six, but that Gneed 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
Received: 13 July 1964
Published: 1 May 1966
Ernest Edward Shult