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A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the generalized order of an element in a finite group $G$ using its character table., Comment: 11 pages

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. In this paper we prove that the derived subgroup $\nu(G)'$ is a central product of three normal subgroups of $\nu(G)$, all isomorphic to the non-abelian tensor square $G \otimes G$. As a consequence, we describe the structure of each term of the derived and lower central series of the group $\nu(G)$.

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 $\omega(G)$. Let $G$ be a virtually nilpotent group such that $\omega(G)< \infty$. We prove that $G = K \rtimes H$ where $H$ is a torsion subgroup and $K$ is a torsion-free nilpotent radicable characteristic subgroup of $G$. Moreover, we prove that $G^{'}= D \times \Tor(G^{'})$ where $D$ is a torsion-free nilpotent radicable characteristic subgroup. In particular, if the maximum normal torsion subgroup $\tau(G)$ of $G$ is trivial, then $G^{'}$ is nilpotent.

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index $p^{\alpha}$ (for some prime $p$), then the quotient $G/\mbox{rad}(G)$ of $G$ over the solvable radical is asymptotically small in comparison to $p^{\alpha}!$ (Theorem 4).

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 $\omega(G)$. We prove that if $G$ is a soluble group with finite rank such that $\omega(G)< \infty$, then $G$ contains a torsion-free characteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $\omega(G)=3$., Comment: Submitted to an international journal Geometriae Dedicata (2020)

Let $n$ be a positive integer and let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. Set $T_{\otimes}(G) = \{g \otimes h \mid g,h \in G\}$. We prove that if the size of the conjugacy class $\left |x^{\nu(G)} \right| \leq n$ for every $x \in T_{\otimes}(G)$, then the second derived subgroup $\nu(G)''$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.

The operator, $\chi $, of weak commutativity between isomorphic groups $G$ and $G^{\varphi }$ was introduced by Sidki as \begin{equation*} \chi (G)=\left\langle G \cup G^{\varphi }\mid \lbrack g,g^{\varphi }]=1\,\forall \,g\in G\right\rangle \text{.} \end{equation*} It is known that the operator $\chi $ preserves group properties such as finiteness, solubility and also nilpotency for finitely generated groups. We prove that if $G$ is a locally finite group with $exp(G)=n$, then $\chi(G)$ is locally finite and has finite $n$-bounded exponent. Further, we examine some finiteness criteria for the subgroup $D(G) = \langle [g_1,g_2^{\varphi}] \mid g_i \in G\rangle \leqslant \chi(G)$ in terms of the set $\{[g_1,g_2^{\varphi}] \mid g_i \in G\}$.

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^{\varphi}]$ and the $k$-th term of the lower central series $\gamma_k(\nu(G))$ are potently embedded in $\nu(G)$ (Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the exponent $\exp(\nu(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study the weak commutativity construction of powerful $p$-groups (Theorem C).

By using finiteness related result of non-abelian tensor product we prove finiteness conditions for the homotopy groups $\pi_n(X)$ in terms of the number of tensors. In particular, we establish a quantitative version of the classical Blakers-Massey triad connectivity theorem. Moreover, we study others finiteness conditions and equivalence properties that arise from the non-abelian tensor square. In the end, we give applications to homotopy pushout, especially in the case of Eilenberg-MacLane spaces.

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $[G,H]$ the derivative subgroup of $G$ under $H$. We prove that if the set $\{g^{-1}g^h \mid g \in G, h \in H\}$ has $m$ elements, then the derivative $[G,H]$ is finite with $m$-bounded order. Moreover, we show that if the set of all tensors $T_{\otimes}(G,H) = \{g\otimes h \mid g \in G, h\in H\}$ has $m$ elements, then the non-abelian tensor product $G \otimes H$ is finite with $m$-bounded order. We also examine some finiteness conditions for the non-abelian tensor square of groups., Comment: 9 pages

Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally virtually nilpotent. Further, let $d,n$ be positive integers and $w$ a non-commutator word. Assume that $G$ is a $d$-generator residually finite group in which all $w$-values are $n$-Engel. We show that the verbal subgroup $w(G)$ has $\{d,n,w\}$-bounded nilpotency class., Comment: 9 pages. arXiv admin note: text overlap with arXiv:1505.04468

Let $G$ be a finite group and let $k \geq 2$. We prove that the coprime subgroup $\gamma_k^*(G)$ is nilpotent if and only if $|xy|=|x||y|$ for any $\gamma_k^*$-commutators $x,y \in G$ of coprime orders (Theorem A). Moreover, we show that the coprime subgroup $\delta_k^*(G)$ is nilpotent if and only if $|ab|=|a||b|$ for any powers of $\delta_k^*$-commutators $a,b\in G$ of coprime orders (Theorem B).

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 journal

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $\lambda(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups $G$ with $\lambda(G)=6$. Moreover, we show that if $G$ is a group with $\lambda(G) \leqslant 30$, then $G$ is solvable., Comment: Submitted to an internacional journal

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)

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.

Let $G$, $H$ be groups. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. We prove that if $G$ and $H$ are groups that act compatibly on each other and such that the set of all tensors $T_{\otimes}(G,H)=\{g\otimes h \, : \, g \in G, \, h\in H\}$ is finite, then the non-abelian tensor product $G \otimes H$ is finite. In the opposite direction we examine certain finiteness conditions of $G$ in terms of similar conditions for the tensor square $G \otimes G$., Comment: The content of this paper improves and extends the main results of arXiv:1603.07003 [math.GR]

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a finitely generated group in which the set of all simple tensors $T_{\otimes}(G)$ is finite, then the non-abelian tensor square $G \otimes G$ and the group $\nu(G)$ are finite. Moreover, we show that if $G$ is a locally residually finite group in which the set of simple tensors $T_{\otimes}(H)$ is finite for every proper finitely generated subgroup $H$ of $G$, then the group $\nu(G)$ is locally finite., Comment: This paper has been withdrawn by the author due to important modifications to be done

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.

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:1505.04468

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.

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).

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 Algebra

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 pages

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.

Let m, n be positive integers, v a multilinear commutator word and w = v^m. 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.