Solvable conjugacy class graph of groups

Research output: Working paper/Preprint › Preprint

Authors

  • Parthajit Bhowal
  • Peter J. Cameron
  • Rajat Kanti Nath
  • Benjamin Sambale

Research Organisations

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Details

Original languageEnglish
Publication statusE-pub ahead of print - 5 Dec 2021

Abstract

In this paper we introduce the graph \(\Gamma_{sc}(G)\) associated with a group \(G\), called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of \(G\) and two distinct conjugacy classes \(C, D\) are adjacent if there exist \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle\) is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~\(d\), and we find explicitly the list of such groups with \(d=2\).

Keywords

    math.CO, math.GR, 05C25

Cite this

Solvable conjugacy class graph of groups. / Bhowal, Parthajit; Cameron, Peter J.; Nath, Rajat Kanti et al.
2021.

Research output: Working paper/Preprint › Preprint

Bhowal, P, Cameron, PJ, Nath, RK & Sambale, B 2021 'Solvable conjugacy class graph of groups'. <http://arxiv.org/abs/2112.02613v3>
Bhowal, P., Cameron, P. J., Nath, R. K., & Sambale, B. (2021). Solvable conjugacy class graph of groups. Advance online publication. http://arxiv.org/abs/2112.02613v3
Bhowal P, Cameron PJ, Nath RK, Sambale B. Solvable conjugacy class graph of groups. 2021 Dec 5. Epub 2021 Dec 5.
Bhowal, Parthajit ; Cameron, Peter J. ; Nath, Rajat Kanti et al. / Solvable conjugacy class graph of groups. 2021.
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AU - Cameron, Peter J.

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N2 - In this paper we introduce the graph \(\Gamma_{sc}(G)\) associated with a group \(G\), called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of \(G\) and two distinct conjugacy classes \(C, D\) are adjacent if there exist \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle\) is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~\(d\), and we find explicitly the list of such groups with \(d=2\).

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