Related papers: A note on FC-nilpotency

A subgroup of a finite group is wide if each prime divisor of the group order divides the subgroup order. We obtain the description of finite soluble groups with no wide subgroups. We also prove that a finite soluble group with nilpotent…

A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954 B. H. Neumann discovered that if G is a BFC-group then the derived group G' is finite. Let w=w(x_1,\dots,x_n) be a multilinear commutator. We study…

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are…

In SGA 2, Grothendieck conjectures that the \'etale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In…

In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a similar approach for finite nilpotent semigroups. This differs from the group theory setting in…

We show how a recent result of Hrushovsky implies that if an asymptotic cone of a finitely generated group is locally compact, then the group is virtually nilpotent.

Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the…

The article deals with profinite groups in which the centralizers are pronilpotent (CN-groups). It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CN-group, and let F be the maximal normal…

A version of Paley-Wiener like theorem for connected, simply connected nilpotent Lie groups is proven.

Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its…

We prove that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight, which was conjectured by Parshin. Note that we consider (possibly, infinite-dimensional)…

We study the action of a nilpotent group G with finite generating set S on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting S into the infinite component of…

We show that, for a finitely generated residually finite group $\Gamma$, the word $[x_1, \ldots, x_k]$ is a probabilistic identity of $\Gamma$ if and only if $\Gamma$ is virtually nilpotent of class less than $k$. Related results,…

We consider finitely generated group endowed with a word metric. The group acts on itself by isometries, which induces an action on its horofunction boundary. The conjecture is that nilpotent groups act trivially on their reduced boundary.…

Let $A$ be a non-metacyclic finite group. Suppose that $A$ acts coprimely on a finite group $G$ in such a manner that $C_G(a)$ is nilpotent for any $a\in A^{\#}$. In the present paper we investigate some conditions on $A$ which imply that…

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group…

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a…

We construct an analogue of Neumann's affiliated algebras for sofic group algebras over arbitrary fields. Consequently, we settle Kaplansky's direct finiteness conjecture for sofic groups.