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Dominating sets in intersection graphs of finite groups. (English) Zbl 1483.20043

Summary: 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 study the dominating sets in intersection graphs of finite groups. We classify abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related to Burnside rings. We introduce the notion of an intersection graph of a \(G\)-set (somewhat generalizing the ordinary definition of an 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 the Burnside ring. The intersection graph of \(G\) is the \(1\)-skeleton of the simplicial complex. We name 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 the domination number of \(\Gamma(G)\) is 1, then the intersection complex of \(G\) is contractible.

MSC:

20D30 Series and lattices of subgroups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
55U10 Simplicial sets and complexes in algebraic topology
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