Remark on subgroup intersection graph of finite abelian groups

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.


Introduction
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]. For example, B. Csákány and G. Pollák [3] defined the intersection graph of a group G, whose vertices are the proper non-trivial subgroups of G, and two vertices H 1 and H 2 are adjacent if ≠ H H 1 2 and they have a non-trivial intersection. In [4], B. Zelinka continued the work on intersection graphs of finite abelian groups. R. Shen classified finite groups with disconnected intersection graphs in [5].
The authors of [6] introduced the power graph of group G, whose vertices set is G and two vertices x and y are adjacent if ≠ x y and one is a power of the other. P. J. Cameron and S. Ghosh [7] proved that finite abelian groups with isomorphic power graphs are isomorphic. They also conjectured that G 1 and G 2 have the same number of elements of each order if their power graphs are isomorphic.
In this paper, we mainly study the subgroup intersection graph of finitely generated abelian groups, which is denoted by ( ) G Γ . This graph was defined by T. T. Chelvam and M. 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 ( ) G Γ in [8]. In particular, they determined that when ( ) G Γ is complete or Eulerian. In [9], T. T. Chelvam and M. Sattanathan obtained all planar ( ) G Γ and unicycle ( ) G Γ for abelian groups. Our main result shows that two finite abelian groups with isomorphic subgroup intersection graphs are isomorphic. We also give example to show that this result is not valid for non-abelian groups and infinite abelian groups. Figure 1 shows the subgroup intersection graph of the cyclic group of order 6. This paper is organized as follows. In Section 2, we introduce notation and terminology of graphs and groups and review some of the standard facts of finite generated abelian group. In Section 3, we determine the structure of subgroup intersection graphs on finite abelian p-groups and show that two finite abelian p-groups are isomorphic if their subgroup intersection graphs are isomorphic. We also give an example to show that the result is not valid for non-abelian groups. In Section 4, we present and prove our main result.

Preliminaries
We follow the notations of graph used in [10]. We will write ( ) V Γ and ( ) E Γ for the set of vertices and the set of edges of Γ. For ∈ ( ) x y V , Γ, the notation ( ) ∈ ( ) x y E , Γ means that x and y are adjacent. The induced subgraph of Γ on a vertex set ⊂ ( ) Γ. Let { } ∈ Γ i i I be a collection of graphs. The union graph of Γ i , denoted by ∪ ∈ Γ i I i , is the graph whose vertex set and edge set are disjoint union of ( ) V Γ i and ( ) E Γ i , respectively. Hence, each graph is the union of its connected components. For simplicity of notation, we write = n Γ Γ 1 if Γ is the union of n copies of Γ . 1 Next, we review some results of the structure of finite generated abelian groups. For convenience, we repeat the relevant material from Chapter I of [11] without proof.
Throughout this paper, a group G is always written multiplicative. The identity of G will be denoted by 1 G and the subset of non-identity elements of G will be denoted by G ⁎ . The order of an element ∈ g G, denoted by ( ) o g , is the minimal positive integer k such that = g 1 k G . If no such k exists, then the order of g is said to be infinite. An element g is called torsion if ( ) < ∞ o g . If G is a finite group, the exponent of G is the least common multiple of order of all elements of G. We will denote by G tor the set of all torsion elements of a group G. If G is abelian, then G tor is a subgroup of G. If G is abelian and p is a prime, we denote by ( ) G p the subgroup of G of all elements whose order is a power of p. A finitely generated abelian torsion group is finite. We next describe the structure of finite abelian p-   3 The structure of ( ) G Γ for a finite p-group We fixed a prime p in this section. In [8], T. T. Chelvam and M. Sattanathan show that the subgroup intersection graph of every finite p-group is a union of complete graphs. We give a total description of finite abelian p-groups.
Note that |{ ∈ = }| = z G z x n :  To the best of our knowledge, it is not a simple task to determine the number of solutions = x a p k in non-abelian groups, see [12,13]. In fact, Theorem 3.1 is not true for non-abelian groups. Let G 1 be the finite abelian p-group of type ( ) p p p , , and let G 2 be the non-abelian group with presentation Then both G 1 and G 2 have exponent p, but Remark on subgroup intersection graph of finite abelian gr oups  1027 An independent set S of a graph Γ is a subset of ( ) V Γ such that u and v are not adjacent for any ∈ u v S , . The independent number of Γ denoted by ( ) β Γ is equal to {| |} S max , where S runs over all independent sets of Γ. Proof. Let = { } ∈ A H i i I be the collection of all cyclic subgroups of prime order of G and let X be a maximal independent set of ( ) G Γ . We claim that: for any given ∈ H A, there exists exactly one ∈ x X such that ⊆ 〈 〉 H x . Suppose that ⊈〈 〉 H x for any ∈ x X. Choose a non-identity element ∈ z H. Since H is a cyclic group of prime order, one has |〈 〉 ∩ 〈 〉| = | ∩ 〈 〉| > z x H x 1 if and only if ⊆ 〈 〉 H x . So z is not adjacent to any point in X and ∪ { } X z is also an independent set. This contradicts to the choice of X. If ⊆ 〈 〉 H x and ⊆ 〈 〉 H y for ≠ x y, then ⊂ 〈 〉 ∩ 〈 〉 H x y and x and y are adjacent. This proves the claim. We define a map → σ A X : x . By the claim, σ is well-defined and surjective. Therefore, is also an independent set with size ( ( )) β G Γ . Now suppose that ∈ a X and ( ) o a is not a power of a prime. Then 〈 〉 a contains at least two distinct subgroups of prime order. So → σ A X : is not injective in this case and | | < | | X A. The proof is finished. □ Suppose that G is a finitely generated abelian group of rank r. Then is the union graph of ( ) G Γ tor and infinite countable copies of ∞ K . . Let p be a prime divisor of n. Since the induced subgraph of Γ i on the subset ( ) G p i ⁎ is isomorphic to ( ( )) G p Γ i , it suffices to prove that ( ( ) ) = ( ) φ G p G p .