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We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines p-nilpotence of the group if a certain condition is met.

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there...

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper, we classify finite solvable groups whose intersection graphs are not $2$-connected and finite nilpotent groups whose intersection graphs are not $3$-connected. Our methods are elementary., Corrected according to the referee's comments. In particular, statements and proofs of Lemma 2 and Lemma 12 are changed. Also the proof of Theorem A in this version is due to I. M. Isaacs

Let G be a group. The intersection graph $${\mathcal{G}(G)}$$ of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and there is an edge between two distinct vertices X and Y if and only if $${X \cap Y \neq 1}$$ where 1 denotes the trivial subgroup of G. It was conjectured in [3] that two (noncyclic) finite abelian groups with isomorphic intersection graphs are isomorphic. In this paper we study this conjecture and show that it is almost true. For any finite abelian group D let D nc be the product of all noncyclic Sylow subgroups of D. Our main result is that: given any two (nontrivial) finite abelian groups A and B, their intersection graphs $${\mathcal{G}(A)}$$ and $${\mathcal{G}(B)}$$ are isomorphic if and only if the groups A nc and B nc are isomorphic, and there is a bijection between the sets of (nontrivial) cyclic Sylow subgroups of A and B satisfying a certain condition. So, in particular, two finite abelian groups with isomorphic intersection graphs will be isomorphic provided that one of the groups has no (nontrivial) cyclic Sylow subgroup. Our methods are elementary.

The intersection graph of a group G is an undirected graphwithout loops and multiple edges defined as follows: the vertex set is theset of all proper non-trivial subgroups of G, and there is an edge betweentwo distinct vertices Hand Kif and only if H∩K6= 1 where 1 denotes thetrivial subgroup of G.In this paper we characterize all finite groups whoseintersection graphs are planar. Our methods are elementary. Among thegraphs similar to the intersection graphs, we may count the subgrouplattice and the subgroup graph of a group, each of whose planarity wasalready considered before in [2, 10, 11, 12]. 1. Introduction and preliminariesA graph is called planar if it can be drawn on the plane in such a way thatits edges intersect only at their endpoints. There are interesting graphs con-structed from algebraic objects such as the subgroup lattice and the subgroupgraph of a group. Planarity of the subgroup lattice and the subgroup graph ofa group were studied by Bohanon and Reid in [2] and by Schmidt in [10, 11]and by Starr and Turner III in [12], and planarity of the intersection graph ofa module over any ring was studied in [13].Here we study planarity of the intersection graph of a finite group. Let G bea group. By the intersection graph of G we mean an undirected graph withoutloops and multiple edges defined as follows: the vertex set is the set of allproper non-trivial subgroups of G, and there is an edge between two distinctvertices H and K if and only if H∩K 6= 1 where 1 denotes the trivial subgroupof G.We call a group planar if its intersection graph is planar. For any naturalnumbers m and n, we use C

Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper we studied the dominating sets in intersection graphs of finite groups. It turns out a subset of the vertex set is a dominating set if and only if the union of the corresponding subgroups contains the union of all minimal subgroups. We classified abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related with Burnside rings. We introduce the notion of intersection graph of a $G$-set (somewhat generalizing the ordinary definition of intersection graph of a group) and establish a general upper bound for the domination number of $\Gamma(G)$ in terms of subgroups satisfying a certain property in Burnside ring. Intersection graph of $G$ is the $1$-skeleton of the simplicial complex whose faces are the sets of proper subgroups which intersect non-trivially. We call this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that if domination number of $\Gamma(G)$ is $1$, then intersection complex of $G$ is contractible.