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 ?(G)is the least positive integer n such that G is the union of the n abelian subgroups.?(G)denotes the order of the maximal subsets of pairwise non-commuting elements.Obviously,?(G)? ?(G).In this paper,we study the group G with ?(G)= 3 and?(G)= 4.In chapter 3,we study the group G with ?(G)= 3 or ?(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 ?(G)= 5.We get the following result.Theorem 0.3 Let G be the finite group,then ?(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.