GROUPS IN WHICH THE NORMAL CLOSURES OF CYCLIC SUBGROUPS HAVE FINITE SECTION RANK

Abstract

A group G is said to have finite section p-rankrp(G) = r (here p is a prime) if every elementary abelian p-section U/V of G is finite of order at most pr and there is an elementary abelian p-section A/B of G such that |A/B| = pr. If ℙ is the set of all primes and λ : ℙ → ℕ ∪ {0} is a function, we say that a group G has λ-bounded section rank if rp(G) ≤ λ(p) for each p ∈ ℙ. In this paper we show that if G is a locally generalized radical group in which the normal closures of the cyclic subgroups of G have finite λ-bounded section rank, then [G,G] has [Formula: see text]-bounded section rank for some function [Formula: see text]. This is a wide generalization of some results by Neumann, Smith and many others. Moreover we are able to give explicit formulas for the involved bounds.