# Subgroup intersection graph of finite abelian groups

Document Type: Research Paper

Authors

Manonmaniam Sundaranar University

Abstract

Let $G$ be a finite group with the identity $e$‎. ‎The subgroup intersection graph $\Gamma_{SI}(G)$ of $G$ is the graph with vertex set $V(\Gamma_{SI}(G)) = G-e$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma_{SI}(G)$ if and only if $|\left\langle x\right\rangle \cap\left\langle y\right\rangle|>1$‎, ‎where $\left\langle x\right\rangle$ is the cyclic subgroup of $G$ generated by $x\in G$‎. ‎In this paper‎, ‎we obtain a lower bound for the independence number of subgroup intersection graph‎. ‎We characterize certain classes of subgroup intersection graphs corresponding to finite abelian groups‎. ‎Finally‎, ‎we characterize groups whose automorphism group is the same as that of its subgroup intersection graph‎.

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