Groups with regular automorphisms of order four

An automorphism of a group G is called regular if it moves every element of G except the identity. BURNSIDE proved that a finite group G has a regular automorphism of order two if and only if G is an abelian group of odd order, and then the only such automorphism maps every element onto its inverse ([21, p. 230). More recently several authors considered the question: what groups can admit regular automorphisms of order a prime p ? B. H. NEU~IANN [8] and M. NAGATA [7] extended the original theorem about the case p = 2 to varioug classes of infinite groups. The case p = 3 appeared in the second edition of BURNSlBE'S book ([81, pp. 90--91), a~ad this theorem was also generalized by NEUMANN [9]. For arbi trary p, the results so far can be summarized as follows. If a locally finite or locally nilpotent group G has a regular, automorphism of prime order p, then G is nilpotent and its class is bounded in terms of p (G. HIGMAN [d] a n d J. THOMPSON [10l). In this paper we consider groups which admit regular automorphisms of order four. For this case one cannot obtain results like those quoted above. Finite nilpotent groups of every class have regular automorphisms of order four; hence locally nilpotent but non-nilpotent groups have them too. There are also non-nilpotent finite soluble groups, both metabelian and nonmetabelian ones, that admit such automorphisms. (Examples are given in the last section of the paper.) What we can prove is that i[ G is either a locally nilpotent or a periodic locally soluble group, and i / G has a regular automorphism o/order ]our, then the ,second derived group o /G is contained in the centre o/ G.*) The examples just mentioned show this conclusion to be in a sense the best p0~sible. All outline of the proof is the following. Let G be an arbi trary group with a regular automorphism ~ of order four, and let us consider the action of 0c 2. I t is clear that 0r ~ need not be regular: the elements of G fixed by 0t 2 form a subgroup T(G). The restriction of to this subgroup is a regular automorphism of order t w o a n d so; under any of the conditions tha t we later adopt, T(G) is abelian and its elements are all inverted by 0r I t is more difficult to see what happens to the elements