FIXED-POINT-FREE OPERATOR GROUPS OF ORDER 8

Let A be a group of order 2n which acts as a fixed-pointfree 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. © 1969 by Pacific Journal of Mathematics.