A defining property of virtually nilpotent groups

Bulletin of the Malaysian Mathematical Sciences Society, 2020

If X is a class of groups, then a group G is called a minimal non-X group if its proper subgroups are X-groups while G itself do not belong to X. In the 1980s, locally graded minimal non-(finite-by-abelian) groups have been characterized, and recently locally graded minimal non-hypercentral groups have been investigated. Here, we prove that a locally graded group is a minimal non-(finite-by-hypercentral) group if, and only if, it is either a minimal non-hypercentral group or a minimal non-(finite-by-abelian) group. Keywords Hypercentral • Finite-by-abelian • Finite-by-hypercentral • Minimal non-X Mathematics Subject Classification 20F19 • 20E07 1 Introduction Let X be a class of groups. A group G is called a minimal non-X group, MNX-group in short, if all its proper subgroups belong to X, while G itself is not in X. The study of MNX-groups started in the beginning of the last century when Miller and Moreno [8] and Schmidt [13] characterized finite MNA-groups and finite MNN-groups, respectively, where A and N denote the class of abelian and nilpotent groups. Among the Communicated by V. Ravichandran.

Journal of the London Mathematical Society, 1968

This article examines aspects of the theory of locally nilpotent linear groups. We also present a new classification result for locally nilpotent linear groups over an arbitrary field F.

Groups, Geometry, and Dynamics, 2014

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Mathematics Subject Classification (2010) 20F16, 20F65. in a series of papers in the late 1990s: in [5], Brookes, Roseblade and Wilson showed that a finitely presented abelian-by-polycyclic group G is necessarily nilpotent-by-nilpotent-byfinite; in then, this conclusion is sharpened to G is nilpotent, by nilpotent of class at most two, by finite.

2006

Bulletin of the Australian Mathematical Society, 2003

Let c ≥ 0, d ≥ 2 be integers and be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that is locally nilpotent? We prove that if c ≤ 2d + 2d−1 − 3 then the variety is locally nilpotent and we reduce the question of Gupta about the periodic groups in to the prime power exponent groups in this variety.

Canadian mathematical bulletin, 1994

Bulletin of the Australian Mathematical Society, 1970

If is a variety of groups which are nilpotent of class c then is generated by its free group of rank c. It is proved that under certain general conditions cannot be generated by its free group of rank c - 2, and that under certain other conditions is generated by its free group of rank c - 1. It follows from these results that if is the variety of all groups which are nilpotent of class c, then the least value of k such that the free group of of rank k generates is c - 1. This extends known results of L.G. Kovács, M.F. Newman, P.P. Pentony (1969) and F. Levin (1970).

Glasgow Mathematical Journal

Abstarct Let γ n = [x1,…,x n ] be the nth lower central word. Denote by X n the set of γ n -values in a group G and suppose that there is a number m such that $|{g^{{X_n}}}| \le m$ for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

In this note we determine explicit formulas for the relative com- mutator of groups with respect to the subvarieties of n-nilpotent groups and of n-solvable groups. In particular these formulas give a characterization of the extensions of groups that are central relatively to these subvarieties.

Journal of the Australian Mathematical Society, 2003

If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.