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ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES
Published online by Cambridge University Press: 12 May 2021
Abstract
Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 101 - 105
- Copyright
- © 2021 Australian Mathematical Publishing Association Inc.
Footnotes
The research of the second author was in part supported by a grant from IPM (No. 1400200028).