Year: 2019
Type: preprint
Abstract: The operator, $χ$, of weak commutativity between isomorphic groups $G$ and $G^{φ}$ was introduced by Sidki as \begin{equation*} χ(G)=\left\langle G \cup G^{φ}\mid \lbrack g,g^{φ}]=1\,\forall \,g\in G\right\rangle \text{.} \end{equation*} It is known that the operator $χ$ 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 $χ(G)$ is locally finite and has finite $n$-bounded exponent. Further, we examine some finiteness criteria for the subgroup $D(G) = \langle [g_1,g_2^φ] \mid g_i \in G\rangle \leqslant χ(G)$ in terms of the set $\{[g_1,g_2^φ] \mid g_i \in G\}$.
Source: arXiv (Cornell University)
Authors: Bastos, Raimundo, Lima, Bruno, Nunes, Ricardo
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2954777585