Finite Groups That Are The Union Of Less Than Six Abelian Subgroups

In finite groups,we study the structure and properties of groups which are the unions of sever abelian subgroups.For a finite group G,the covering number α(G)is the least positive integer n such that G is the union of the n abelian subgroups.ω(G)denotes the order of the maximal subsets of pairwise non-commuting elements.Obviously,α(G)≤ ω(G).In this paper,we study the group G with α(G)= 3 andα(G)= 4.In chapter 3,we study the group G with α(G)= 3 or α(G)= 4,and get the following results.Theorem 0.1 Let G be the finite group,then the following results hold:G is the union of three abelian subgroups if and only if G/Z(G)≌ Z2 × Z2.Theorem 0.2 Let G be the finite group,then the following results hold:G is the union of four abelian subgroups if and only if G/Z(G)≌ S3 or G/Z(G)≌ Z3 × Z3.In chapter 4,we study the nilpotent group G with α(G)= 5.We get the following result.Theorem 0.3 Let G be the finite group,then α(G)= 5 if and only if one of the following results holds:(1)if G is nonnilpotent,then G/Z(G)≌ A4;(2)if G is nilpotent,then G/Z(G)≌ C2 × C2 × C2 × C2 or G/Z(G)≌ D8.