Nilpotency in Groups with Chain Conditions

IN this paper we shall study the connection between local and global nilpotency for groups which satisfy certain chain conditions. In the first section, we shall prove that a group satisfying the chain condition on centralizers has a unique maximal normal nilpotent subgroup (the Fitting subgroup), i.e. the group generated by all normal nilpotent subgroups must again be nilpotent. This generalizes results of Bryant [1] for the periodic case, and Poizat-Wagner [6] and Wagner [8] for the case of stable groups. We shall then use a result of Gruenberg [3] to deduce a characterisation of Engel elements. In the second section, we shall consider a stronger chain condition, namely on intersections of families of uniformly relatively definable subgroups. Although it sounds unwieldy, it occurs naturally in model theory in the analysis of stable or substable groups. In particular, it is satisfied by linear groups. The main result here states that a locally nilpotent group satisfying this chain condition must be hypercentral. We then apply this to characterize left Engel elements in terms of the Fitting subgroup and the Hirsch-Plotkin radical, and give partial information about the hypercentrality of right Engel elements, generalizing results from [7]. Notation. Our commutators are left-normalized, defined inductively via [g0, gu ..., gn+i] = [\g0, gi,.-., gn], gn+i]', repeated commutators are given by [g,oh] = g and [g,n+\h] = [gn h], h]. The derived series is denoted by {G}, the descending central series by yn(G), and the ascending central series by {Zn(G)}\ the last one may be continued transfinitely by taking unions at the limit stages. The hypercentre ZJJ3) is the union of all iterated centres, and G is called hypercentral if G = Za(G) for some ordinal a. A group is locally nilpotent if any finitely generated subgroup is nilpotent; it is uniformly locally nilpotent if any finitely generated subgroup is nilpotent and the nilpotency class depends only on the number of generators. The series of iterated centralizers of A in G modulo H is defined inductively via C%{AIH) = H and Cc \A/H) = Cc{AICc(AIH)r\C\Nc{C'c{AIH), again taking unions at limit ordinals. Note that Za{G) = Cc(G) f° r a n v Finally, if a group G acts on a group H, we shall talk about centralizers as if in H X G.