Subgroup intersection graph of finite abelian groups

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 $|leftlangle xrightrangle capleftlangle yrightrangle|>1$‎, ‎where $leftlangle xrightrangle $ is the cyclic subgroup of $G$ generated by $xin G$‎. ‎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.