Domination number of a finite group

Abstract

In this thesis we determine the domination number of a Dihedral group. Given a finite group G one may construct a graph whose vertices are proper subgroups of G where a proper subgroup of G is a subgroup different than 1 and G. In this graph there is an edge between two distinct vertices if the intersection of corresponding subgroups is not 1. By the domination number of G we mean the domination number of that graph. Let G be a finite group. A set A of proper subgroups of G is called a dominating for G if for any proper subgroup H of G there is an element X of A such that H intersection X not equal to 1. Among all the dominating sets for G as G is finite, there must be a dominating set whose cardinality is the smallest. The cardinality of such a dominating set is called the domination number of G. Similar graphs constructed from rings, vector spaces, abelian groups, and modules were studied in many papers. However the domination number of a graph constructed from a finite group has not been determined yet. We determine the domination number of a Dihedral Group which is a non-abelian finite group.