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.
Highlights
There are many papers on assigning a graph to a group
The most famous example is the Cayley graphs, whose vertices are elements of groups and adjacency relations are defined by subsets of the groups, see [1,2]
We mainly study the subgroup intersection graph of finitely generated abelian groups, which is denoted by Γ(G)
Summary
There are many papers on assigning a graph to a group. The most famous example is the Cayley graphs, whose vertices are elements of groups and adjacency relations are defined by subsets of the groups, see [1,2]. We mainly study the subgroup intersection graph of finitely generated abelian groups, which is denoted by Γ(G) Sattanathan in [8] as follows: the vertices are the non-identity elements of G, and two vertices x and y are adjacent if and only if x ≠ y and |〈x〉 ∩ 〈y〉| > 1, where 〈x〉 is the cyclic subgroup of G generated by x. They characterized some fundamental properties of Γ(G) in [8].
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