Dominating Sets in Intersection Graphs of Finite Groups

TL;DR

This paper explores the properties of dominating sets in intersection graphs of finite groups, classifies abelian groups by domination number, and relates these graphs to algebraic structures like Burnside rings and topological complexes.

Contribution

It introduces a new perspective on intersection graphs of finite groups, classifies abelian groups by domination number, and connects these graphs to algebraic and topological concepts.

Findings

A subset is dominating iff its union contains all minimal subgroups.

Classified abelian groups by their domination number.

Established bounds for domination number in specific group classes.

Abstract

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.