Fixed-point-free operator groups of order 8

Fletcher Gross

Vol. 28 (1969), No. 2, 357–361

DOI: 10.2140/pjm.1969.28.357

Abstract

Let A be a group of order 2ⁿ which acts as a fixed-point-free group of operators on the finite solvable group G. If no additional assumptions are made concerning G, then “reasonable” upper bounds on the nilpotent length, l(G), of G have been obtained only when A is cyclic [Gross] or elementary abelian [Shult]. As a small step in extending the class of 2-groups A for which such bounds exist, it is shown in the present paper that if |A| = 8, then l(G) ≦ 3 if A is elementary abelian or quaternion and l(G) ≦ 4 otherwise.

Mathematical Subject Classification

Primary: 20.40

Milestones

Received: 19 March 1968

Published: 1 February 1969