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.

The metanorm, a characteristic subgroup: Embedding properties

Thenormof a group was introduced by R. Baer as the intersection of all normalizers of subgroups, and it was later proved that the norm is always contained in the second term of the upper central series of the group. The aim of this paper is to study embedding properties of themetanormof a group, defined as the intersection of all normalizers of non-abelian subgroups. The metanorm is related to the so-calledmetahamiltonian groups, i.e. groups in which all non-abelian subgroups are normal, and it is known that every locally graded metahamiltonian group is finite over its second centre. Among other results, it is proved here that ifGis a locally graded group whose metanormMis not nilpotent, then{M^{\prime}/M^{\prime\prime}}is a small eccentric chief factor and it is the only obstruction to a strong hypercentral embedding ofMinG.

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.

Infinite minimal non-hypercyclic groups

If 𝔛 is a class of groups, a group G is minimal non-𝔛 if it is not an 𝔛-group, but all its proper subgroups belong to 𝔛. The aim of this paper is to prove that for an infinite locally graded group, the property of being minimal non-hypercentral and that of being minimal non-hypercyclic are equivalent. Moreover, the main properties of infinite minimal non-hypercentral groups are described. In the last section, we study groups of infinite rank in which all proper subgroups of infinite rank satisfy a generalized supersolubility condition.

On groups with finitely many derived subgroups

In this paper, we show that every locally graded group with a finite number k of derived subgroups is nilpotent-by-abelian-by-(finite of order ≤ k!)-by-abelian. Also we give some sufficient conditions on the number of derived subgroups of a locally graded group to be solvable.

ON GROUPS ADMITTING A WORD WHOSE VALUES ARE ENGEL

Let m, n be positive integers, v a multilinear commutator word and w = vm. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

Let G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.