Groups with restricted non-permutable subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be permutable if [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text] and the group [Formula: see text] is called metaquasihamiltonian if all subgroups of [Formula: see text] are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group [Formula: see text] is soluble with derived length at most [Formula: see text] and contains a finite normal subgroup [Formula: see text] such that all subgroups of the factor [Formula: see text] are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian.

Groups with an Engel restriction on proper subgroups of infinite rank

We prove that a locally graded group whose proper subgroups are Engel (respectively, [Formula: see text]-Engel) is either Engel (respectively, [Formula: see text]-Engel) or finite. We also prove that a group of infinite rank whose proper subgroups of infinite rank are Engel (respectively, [Formula: see text]-Engel) is itself Engel (respectively, [Formula: see text]-Engel), provided that [Formula: see text] belongs to the Černikov class [Formula: see text], which is the closure of the class of periodic locally graded groups by the closure operations Ṕ, P̀, R and L.

Locally Graded n-Bell Groups

Let n ≠ 0, 1 be an integer and [Formula: see text] be the variety of n-Bell groups defined by the law [xn,y][x,yn]-1 = 1. Let [Formula: see text] be the class of groups in which for any infinite subsets X and Y there exist x ∈ X and y ∈ Y such that [xn,y][x,yn]-1 = 1. In this paper we prove [Formula: see text], where [Formula: see text] and [Formula: see text] are the classes of all finite groups and all locally graded groups, respectively.

On locally graded groups with a word whose values are Engel

Let m, n be positive integers, let υ be a multilinear commutator word and let w = υm. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

Groups with Finiteness Conditions on the Lower Central Series of Subgroups

Let k be a positive integer. Locally graded groups G for which one of the sets {γk(H)| H ≤ G} and {γk(H)| H ≤G, H infinite } is finite are classified.

ON GROUPS WITH ALL SUBGROUPS SUBNORMAL OR SOLUBLE OF BOUNDED DERIVED LENGTH

In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.