A study on covering number of finite groups

Abstract

This thesis presents a detailed investigation into the covering number ω(G) of finite groups. A covering of a group G is defined as a collection of proper subgroups whose union equals G, and theminimal such number is called the covering number. We begin by discussing foundational results, such as the non-existence of a covering for cyclic groups and the impossibility of a covering number of two for any group. Special focus is given to the characterization of groups with covering number three, including a detailed structural proof that such groups G satisfy G/K →= C2 ↑ C2 for some normal subgroup K [4]. Furthermore, we discuss computation of exact covering numbers for symmetric groups Sn with odd n > 1, we discuss the proof of the result ω(Sn) = 2n→1, except for the case n = 9 as in [3].