arXiv subjects

Noraí Romeu Rocco

Publications and source records attributed to Noraí Romeu Rocco.

3 recordsLinked to original sources

arXiv2017

A Polycyclic Presentation for the q-Tensor Square of a Polycyclic Group

Let $G$ be a group and $q$ a non-negative integer. We denote by $ν^q(G)$ a certain extension of the $q$-tensor square $G \otimes^q G$ by $G \times G$. In this paper we derive a polycyclic presentation for $G \otimes^q G$, when $G$ is polycyclic, via its embedding into $ν^q(G)$. Furthermore, we derive presentations for the $q$-exterior square $G \wedge^q G$ and for the second homology group $H_2(G, \mathbb{Z}_q).$ Additionally, we establish a criterion for computing the $q-$exterior centre $Z_q^\wedge (G)$ of a polycyclic group $G, $ which is helpful for deciding whether $G$ is capable modulo $q$. These results extend to all $q \geq 0$ existing methods due to Eick and Nickel for the case $q = 0$.

math.GR↗

arXiv2016

A note on finiteness conditions for the non-abelian tensor square of groups

Let $G$ be a group. We denote by $ν(G)$ 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 $ν(G)$ 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 $ν(G)$ is locally finite.

math.GR↗

arXiv2016

The non-abelian tensor square of residually finite groups

Let $m,n$ be positive integers and $p$ a prime. We denote by $ν(G)$ 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 $[x,y^φ]^q = 1$, then the derived subgroup $ν(G)'$ 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 $[x,y^φ]^q$ is left $n$-Engel, then the non-abelian tensor square $G \otimes G$ is locally virtually nilpotent (Theorem B).

math.GR↗