On Independence Number of Comaximal Subgroup Graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918045483532288 |
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| author | Das, Angsuman Mandal, Arnab |
| author_facet | Das, Angsuman Mandal, Arnab |
| contents | In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph $Γ(G)$ that guarantee solvability, supersolvability, and nilpotency of the underlying group $G$. Specifically: \begin{itemize}
\item For solvability, we prove that any group $G$ with independence number $α(Γ(G))\leq 51$ must be solvable, and show that the alternating group $A_5$ is uniquely determined by its graph.
\item For supersolvability, we show that $α(Γ(G))\leq 14$ implies $G$ is supersolvable, except for three explicit exceptions.
\item For nilpotency, we prove that $α(Γ(G))\leq 6$ ensures nilpotency, except for five groups. \end{itemize} Finally, we conclude with some open issues involving domination parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Independence Number of Comaximal Subgroup Graph Das, Angsuman Mandal, Arnab Group Theory 05C25, 20D10, 20D15 In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph $Γ(G)$ that guarantee solvability, supersolvability, and nilpotency of the underlying group $G$. Specifically: \begin{itemize} \item For solvability, we prove that any group $G$ with independence number $α(Γ(G))\leq 51$ must be solvable, and show that the alternating group $A_5$ is uniquely determined by its graph. \item For supersolvability, we show that $α(Γ(G))\leq 14$ implies $G$ is supersolvable, except for three explicit exceptions. \item For nilpotency, we prove that $α(Γ(G))\leq 6$ ensures nilpotency, except for five groups. \end{itemize} Finally, we conclude with some open issues involving domination parameters. |
| title | On Independence Number of Comaximal Subgroup Graph |
| topic | Group Theory 05C25, 20D10, 20D15 |
| url | https://arxiv.org/abs/2506.03848 |