On residually finite groups with Engel-like conditions

Let $m,n$ be positive integers. Suppose that $G$ is a residually finite group in which for every element $x \in G$ there exists a positive integer $q=q(x) \leqslant m$ such that $x^q$ is $n$-Engel. We show that $G$ is locally virtually nilpotent. Further, let $w$ be a multilinear commutator and $G$ a residually finite group in which for every product of at most $896$ $w$-values $x$ there exists a positive integer $q=q(x)$ dividing $m$ such that $x^q$ is $n$-Engel. Then $w(G)$ is locally virtually nilpotent.Comment: 9 page

A Sufficient Condition for Nilpotency of the Commutator Subgroup

Let $G$ be a finite group with the property that if $a,b$ are commutators of coprime orders, then $|ab|=|a||b|$. We show that $G'$ is nilpotent

On profinite groups with Engel-like conditions

Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated profinite group in which for every $\gamma_k$-value $x\in G$ there exists a natural $p$-power $q=q(x)$ such that $x^q$ is Engel. We show that $\gamma_k(G)$ is locally virtually nilpotent

Non-abelian tensor product of residually finite groups

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Suppose that $G$ is residually finite and the subgroup $[G,H] = \langle g^{-1}g^h \ \mid g \in G, h\in H\rangle$ satisfies some non-trivial identity $f \equiv~1$. We prove that if $p$ is a prime and every tensor has $p$-power order, then the non-abelian tensor product $G \otimes H$ is locally finite. Further, we show that if $n$ is a positive integer and every tensor is left $n$-Engel in $\eta(G,H)$, then the non-abelian tensor product $G \otimes H$ is locally nilpotent. The content of this paper extend some results concerning the non-abelian tensor square $G \otimes G$.Comment: Dedicated to Professor Antonio Paques on the occasion of his 70th anniversary, S\~ao Paulo J. Math. Sci. (2017

Non-abelian tensor square of finite-by-nilpotent groups

Let $G$ be a group. We denote by $\nu(G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is finite-by-nilpotent, then the non-abelian tensor square $G \otimes G$ is finite-by-nilpotent. Moreover, $\nu(G)$ is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of $\nu(G)$ (Theorem B)

On finite groups with few automorphism orbits

Denote by $\omega(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $\omega(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $\omega(G) = 6$ and show that if $G$ is a nonsolvable finite group with $\omega(G) = 6$, then either $G \simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.Comment: accepted for publication in Communications in Algebr

The non-abelian tensor square of residually finite groups

Let $m,n$ be positive integers and $p$ a prime. We denote by $\nu(G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a residually finite group satisfying some non-trivial identity $f \equiv~1$ and for every $x,y \in G$ there exists a $p$-power $q=q(x,y)$ such that $[x,y^{\varphi}]^q = 1$, then the derived subgroup $\nu(G)'$ is locally finite (Theorem A). Moreover, we show that if $G$ is a residually finite group in which for every $x,y \in G$ there exists a $p$-power $q=q(x,y)$ dividing $p^m$ such that $[x,y^{\varphi}]^q$ is left $n$-Engel, then the non-abelian tensor square $G \otimes G$ is locally virtually nilpotent (Theorem B).Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1505.0446

A criterion for metanilpotency of a finite group

We prove that the $k$th term of the lower central series of a finite group $G$ is nilpotent if and only if $|ab|=|a||b|$ for any $\gamma_k$-commutators $a,b\in G$ of coprime orders

FC-groups with finitely many automorphism orbits

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. In this paper we prove that if $G$ is an FC-group with finitely many automorphism orbits, then the derived subgroup $G'$ is finite and $G$ admits a decomposition $G = Tor(G) \times D$, where $Tor(G)$ is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup of $Z(G)$. We also show that if $G$ is an infinite FC-group with $\omega(G) \leqslant 8$, then either $G$ is soluble or $G \cong A_5 \times H$, where $H$ is an infinite abelian group with $\omega(H)=2$. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits.Comment: Submitted to an internacional journa

Finite Groups with 6 or 7 Automorphism Orbits

Let $G$ be a group. The orbits of the natural action of \mbox{Aut}(G) on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. In this paper the finite nonsolvable groups $G$ with $\omega(G) \leq 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $\omega(G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $\omega(G) \leq n$, generalizing a result of Kohl