The Maximum Number Of Covers Of Finite Groups

Let G be a group.The subgroups H1,H2,…,Hn,n?2,form a covering of G if(?)Hi=G.A covering is said to be irredundant if none of the subgroups can be removed,i.e.(?)Hj,for all i=1,2,…,n.In this paper,we study the maximum number of the irredundant covering of the group G,which is denoted by ?(G).In this paper,the maximum number of covers A(G)for A1-group,C2-group,finite p-groups of order p4,and the groups of order no more than 36 are calculated.In addition,we classified the finite groups with ?(G)=6 and ?(G)=|G|-t(t?5).This paper is divided into five chapters.In Chapter 1,we introduce the research background,research significance,research methods and main results of this paper.The concepts and lemmas used in this paper are given in Chapter 2.In Chapter 3,we caululate ?(G)of certain groups.It is divided into four sections.the maximum number ?(G)of covers for A1-groups,C2-groups,finite p-groups of order p4,and the groups of order no more than 36 are given,respectively.In Chapter 4,we give the classification of finite groups with certain ?(G).It is divided into two sections.In Section 4.1,we classified the finite groups G with ?(G)=6;In Section 4.2,the finite group G with A(G)=|G|-t(t?5)are classified.In Chapter 5,we show the summary and prospect and explore some problems that can be resolved.