The minimal nilpotent cover of the finite symmetric, alternating and dihedral groups

Abstract

Given a group G, a minimal nilpotent cover is a family of subgroups of G of minimal size subject to the property that all of the subgroups are nilpotent and their union is equal to the group G. A maximal non-nilpotent subset is a subset of G of largest size such that for any two distinct elements, the subgroup they generate is not nilpotent. There has been e ort to investigate covers of groups with particular structure, speci cally normal and abelian covers; much has not been done on nilpotent covers. The main objective of this study was to conduct an analysis of minimal nilpotent cover for each of the three families of groups; nite symmetric group Sn, alternating group An and dihedral group D2k, which are ubiquitous throughout the discipline of mathematics. The speci c objectives were to; investigate nilpotent covers, analyse their relationship with non-nilpotent sub sets, investigate the size of minimal nilpotent cover and maximal non-nilpotent subset and apply group theoretic properties in determining a general formula for the size of the minimal nilpotent cover for each family. The analysis was carried out by means of mathematical proofs based on logical approach together with results from computer algebra package GAP and properties of permutations as well as group action on a k-gon . It has been established that except for A9, the size of minimal nilpotent cover for each of Sn, An for n = 34 10, and D2k, coincide with size of maximal non-nilpotent subset. For D2k, a general formula is produced for k = 34 . The results of this study bene t scientists in identi fying redundancy in the analysis of possible organic molecular structures, solve molecular conformation problems as well as mathematicians in expanding knowl edge in group theoretic concepts. For Sn, only for n = 34 10 were analyzed since as n grows large, it becomes complex and even computer algebra package GAPcannot generate any results. We therefore recommend an investigation on a general formula for minimal nilpotent cover of the nite symmetric group Sn as well as the nite alternating group, An