Groups of finite non-Abelian sectional rank

Abstract

We study non-Abelian locally finite groups and non-Abelian locally solvable groups of finite non-Abelian sectional rank and prove that their (special) rank is finite.The notion of non-Abelian sectional rank of a group was introduced in [1]. In what follows, a section of a group G is understood as a quotient group A/B, where A and B are nonidentity subgroups of the group G, and the subgroup B is normal in A. Recall that a non-Abelian sectional rank of a non-Abelian group G is the minimum number r for which any non-Abelian finitely generated section of the group G can be generated by at most r elements. If all sections of the non-Abelian group G are Abelian, the non-Abelian sectional rank of the group G is assumed to be equal to zero. If G has at least one non-Abelian section and the number r with the indicated properties does not exist, then the non-Abelian sectional rank of the group G is regarded as infinite. We use the notation ~c(G) for the non-Abelian sectional rank of the group G. As is customary, by symbols r(G) and r 0 (G) we denote, respectively, the special rank and the 0-rank of the group G.The finiteness of the non-Abelian sectional rank of a group results in the finiteness of its (special) rank in the classes of non-Abelian locally nilpotent groups [1] and non-Abelian solvable groups [2]. The aim of the present work is to prove a similar assertion for a non-Abelian locally finite group and a non-Abelian locally solvable group. Theorem 1. A non-Abelian locally finite group of finite non-Abelian sectional rank has a finite (special) rank.