This paper considers the question as to which extra conditions on such a group G ensure that G has all subgroups subnormal. For example, if G is torsion-free and locally soluble-by-finite then either G has finite 0-rank or G is nilpotent. | Turk J Math 28 (2004) , 165 – 176. ¨ ITAK ˙ c TUB Groups with Rank Restrictions on Non-Subnormal Subgroups Leonid A. Kurdachenko, Howard Smith Abstract Let G be a group in which every non-subnormal subgroup has finite rank. This paper considers the question as to which extra conditions on such a group G ensure that G has all subgroups subnormal. For example, if G is torsion-free and locally soluble-by-finite then either G has finite 0-rank or G is nilpotent. Several results are obtained on soluble (respectively, locally soluble-by-finite) groups satisfying the stated hypothesis on subgroups. Key Words: Subnormal subgroups; locally soluble-by-finite groups; finite Mal’cev rank. 1. Introduction Let G be a group in which every non-subnormal subgroup has finite rank. Throughout this paper the term “finite rank” means “finite Pr¨ ufer (or Mal’cev, or special) rank”: a group X has finite rank r if every finitely generated subgroup of X is r-generated. It was shown in [5] that if G is soluble and of infinite rank then G is a Baer group, that is, every finitely generated subgroup of G is subnormal, and in [6] it was established that a locally soluble-by-finite group with this restriction on non-subnormal subgroups is soluble (and hence a Baer group). The aim of this article is to present some results on groups in which all non-subnormal subgroups have finiteness of rank of a different kind. We need the following definitions. Let G be a group. (a) G has finite torsion-free rank, or finite 0-rank, denoted r0 (G), if G has a finite subnormal series of subgroups the factors of which 165 KURDACHENKO, SMITH are either infinite cyclic or periodic. (b) For a given prime p, G has finite section p-rank if every elementary abelian p-section of G is finite, and finite section rank if every abelian section has both finite p-rank for every prime p and finite 0-rank. (c) G has finite section total rank if, for each abelian section X of G, r0 (X) + Σrp (X) is finite, where the .
