Groups in which normal closures of elements have boundedly finite rank

A well-known result due to B. H. Neumann states that a group G in which every element has at most n conjugates, where n is a fixed positive integer, has its derived subgroup G’ finite. Subsequent authors discussed improved bounds for the order of G’ in terms of n. On the other hand, it is obvious that if G’ has finite order k then every element of G has at most k conjugates. In 1999 H. Smith studied a related property, the hypothesis there being that the normal closure of every element of G has (Prufer, or Mal'cev) rank at most r, where r is a fixed positive integer. This means that, for every element x of the group G, all finitely generated subgroups of the normal closure <x>^G are r-generated. He proved that a locally (soluble-by-finite) group G satisfying this hypothesis on normal closures has its derived subgroup of r-bounded rank, that is, G’ has finite rank at most s for some integer s that depends on r only. The purpose of this note is to show that the condition on normal closures is in itself sufficient to establish the finiteness of the rank of G’. In fact, it is proved that if the normal closure of every element of a group G has rank at most r then the derived subgroup of G has r-bounded rank.