Subgroups defining automorphisms in locally nilpotent groups

Abstract

We investigate some situation in which automorphisms of a groupG are unique- ly determined by their restrictions to a proper subgroup H. Much of the paper is devoted to studying under which additional hypotheses this property forces G to be nilpotent if H is. As an application we prove that certain countably infinite locally nilpotent groups have uncountably many (outer) automorphisms. Homomorphisms of groups are defined by their restriction to any generating set of their domain. This property actually characterizes generating subsets of groups, for if H is a proper subgroup of the group G then there are two different homomorphisms from G to the same group K whose restrictions to H are the same: a simple construction due to Eilenberg and Moore is suggested as Exercise 3.35 in (10), p.54. The situation can be quite different if—rather than referring to all homomorphisms of domain a group G—we restrict attention to, say, endomorphisms of G only. For instance, if G is isomorphic to the rational group Q and g is any nontrivial element of G, then two endomorphisms of G coincide if they agree on g, in other words endomorphisms of G are uniquely determined by their restrictions to {g}. This suggests the following definition. Let G be a group and let be a set of endomorphisms of G. We say that a subset X of G is a -basis if and only if, for every �,� ∈ , it holds � = � if �|X = �|X, where �|X and �|X denote restrictions to X. We shall also use expressions like 'End- basis', 'Aut-basis' or 'Inn-basis of G' as synonym with EndG-, AutG-, or InnG-basis respectively. For instance, the above example can be rephrased by saying that in the rational group every one- element subset different from the identity subgroup is an End-basis. More generally, every maximal independent subset of a torsion-free abelian group A is an End-basis of A. The Aut-bases of a group G are just the bases of AutG viewed as a permutation group on G, whence the name. Indeed, it is clear that for any ≤ AutG a subset X of G is a -basis if and only if C (X) = 1. In particular, the Inn-bases of a group G are the subsets X of G such that CG(X) = Z(G). Other self-evident facts about -bases (for a set of endomorphisms of a group G) are that if X is a -basis then X1 is a 1-basis for any subset 1 of and any superset X1 of X contained in G. Also, X is a -basis if and only if h Xi is a -basis, so the property of being a -basis could be equivalently defined as an embedding property for subgroups. In this paper we shall assume this point of view and discuss some instances of the general problem of when a group G inherits group theoretical properties from a subgroup of G which is a -basis, for some specific ⊆ EndG. For example, it is immediate to observe that a group is abelian if it has an abelian subgroup which is an Inn-basis (Lemma 1.7). We will be mainly concerned with the case = AutG. A drastically restrictive result is that a direct power of every centreless group can be embedded as a normal subgroup which is an Aut-basis in a group with rather arbitrary structure (see Corollary 1.5). This is the reason why we turn our attention to group classes without nontrivial centreless groups, and mainly study nilpotent (sub)groups.