On the Structure of Finite Groups Associated to Regular Non-Centralizer Graph

Tariq A. Alraqad; Hicham Saber

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arXiv · 1812.09363

On the Structure of Finite Groups Associated to Regular Non-Centralizer Graph

Tariq A. Alraqad·Hicham Saber

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Abstract

The non-centralizer graph of a finite group $G$ is the simple graph $\Upsilon_G$ whose vertices are the elements of $G$ with two vertices $x$ and $y$ are adjacent if their centralizers are distinct. The induced subgroup of $\Upsilon_G$ associated with the vertex set $G\setminus Z(G)$ is called the induced non-centralizer graph of $G$. The notions of non-centralizer and induced non-centralizer graphs were introduced by Tolue in \cite{to15}. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups as well as induced regular groups. Among the many obtained results, we prove that if a group $G$ is regular (resp. induced regular) then $G/Z(G)$ as an elementary $2-$group (resp. $p-$group).

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Tariq A. Alraqad, Hicham Saber. 2018-12-21. On the Structure of Finite Groups Associated to Regular Non-Centralizer Graph. https://arxiv.org/abs/1812.09363

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