On Independence Number of Comaximal Subgroup Graph

TL;DR

This paper establishes specific thresholds on the independence number of the comaximal subgroup graph that determine whether a finite group is solvable, supersolvable, or nilpotent, with some exceptions and open problems.

Contribution

It provides sharp bounds on the independence number of the comaximal subgroup graph that guarantee certain group properties, identifying unique cases and exceptions.

Findings

Groups with independence number ≤ 51 are solvable.

Groups with independence number ≤ 14 are supersolvable, with three exceptions.

Groups with independence number ≤ 6 are nilpotent, with five exceptions.

Abstract

In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph $\Gamma(G)$ 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 $\alpha(\Gamma(G))\leq 51$ must be solvable, and show that the alternating group $A_5$ is uniquely determined by its graph. \item For supersolvability, we show that $\alpha(\Gamma(G))\leq 14$ implies $G$ is supersolvable, except for three explicit exceptions. \item For nilpotency, we prove that $\alpha(\Gamma(G))\leq 6$ ensures nilpotency, except for five groups. \end{itemize} Finally, we conclude with some open issues involving domination parameters.