Remark on subgroup intersection graph of finite abelian groups

Abstract

Abstract Let G be a finite group. The subgroup intersection graph Γ ( G ) \text{Γ}(G) of G is a graph whose vertices are non-identity elements of G and two distinct vertices x and y are adjacent if and only if | 〈 x 〉 ∩ 〈 y 〉 | > 1 |\langle x\rangle \cap \langle y\rangle |\gt 1 , where 〈 x 〉 \langle x\rangle is the cyclic subgroup of G generated by x. In this paper, we show that two finite abelian groups are isomorphic if and only if their subgroup intersection graphs are isomorphic.