Nilpotent height of finite groups admitting fixed-point-free automorphisms

Abstract

w 1. Introduction In this paper, "group" is to mean finite group. We shall consider certain properties of groups admitting fixed-point-free automorphisms (that is, automorphisms which leave only the identity element fixed). Such groups have been studied off and on for many years. BURNSIDE proved before 1897 that a group admitting a fixed-point-free automorphism of period two is abelian, and FRo~ENItJS proved in 1901 that a group admitting one of period three is nilpotent of class at most two (see [2]). The conjecture of FROBENIUS that a group admitting a fixed-point-free automorphism of prime period is nilpotent was proved for solvable groups by WITT about 1936 (unpublished); that the class of nilpotence is bounded by a function of the period was proved by G. I~IGMAN [9] in 1957. The proof of the FROBENIUS conjecture was completed by THOMVSON [I4] in 1959, when he showed that a group admitting a fixedpoint-free automorphism of prime period is solvable. In 1961, GOPCE~STEIN and HERSTEIN [6] proved that a group admitting a fixed-point-free automorphism of period four is solvable and has a nilpotent commutator subgroup. The question of the solvability of groups admitting fixed-point-free automorphisms of arbitrary order has not been settled, and will not be studied here, where we shall assume the groups to be solvable. (Because of the theorem of THOMVSON and FEIT [4], that all groups of odd order are solvable, this restriction is not very severe.) The main result of this paper is a proof that, except possibly in certain special cases, a solvable group admitting a fixed-point-free automorphism of period pn, p a prime, has nilpotent height at most n (see p. 265). We now introduce some of the properties studied in this paper. Let G be a solvable group. Let V(G) be the maximal nilpotent normal subgroup of G. Because the product of two nilpotent normal subgroups is also a nilpotent normal subgroup, V(G) is a weU-defined characteristic subgroup of G. We let V 1 = V(G), and inductively define Vk SO that Vk/Vk_ 1 = V(G/V~_ 1)" V(G) is called the Fitting subgroup of G. If V, = G, the Fitting length or nilpotent height of G is =n. We note that if G and H are groups, V(G x H)= V(G) x V(H). Thus we see easily that the Fitting length of the direct product of two groups is the maximum of the Fitting lengths of the factors. We further note that if 1 c Ni N z ~... ~N~ = G is a characteristic series for G with nilpotent factor groups,