Lower Bounds On The Number Of Maximal Subgroups

Group theory is one of the brilliant mathematical achievements in nineteenth century.On the one hand,it reclaims brand-new fields and becomes the cornerstone of other algebraic structures.On the other hand,its symmetry plays an important role on multiple branches of science,such as crystal structure.Especially,investigating the structure of finite groups is a classical problem,with many open questions remaining.Since the properties of maximal subgroups are closely related to the structure of groups,it is feasible to study the structure of finite groups by maximal subgroups.Many group theorists have made important achieve-ments in this subject,such as the number of conjugate classes of maximal subgroups,the number of weaken second maximal subgroups,and the structure on finite groups in which non-maximal commutative subgroups are with particular properties,etc.In this paper,we give some lower bounds on the number of(non-nilpotent)maximal subgroups of finite non-nilpotent groups.And then we discuss finite p-group,in which all non-maximal abelian subgroups are TI-subgroups.This paper is divided into three chapters.The first chapter mainly introduces the important research results on the number of maximal subgroups and non-maximal abelian subgroups,and give some basic concepts and lemmas used in the next chapters.The second chapter is divided into two sections.The influence of non-normal Sylow sub-groups on the number of(non-nilpotent)maximal subgroups of finite non-nilpotent groups is studied respectively.The following theorems and examples are obtained:Theorem 2.1.1 For a finite group G,let N denote a non-abelian minimal normal subgroup of G and p∈π(N)denote the biggest prime that divides |N|.Assume N is of form N=R1×R2××x Rl,then there are at least pl non-nilpotent maximal subgroups of G which do not contain N,where Ri≌Rj for all i,j∈1,2...,l.Theorem 2.1.2 Let G be a finite non-solvable group.Then we conclude that |n(G)|≥|π(G)|+p,where p is the smallest prime such that P∈Sylp(G)is non-normal.Theorem 2.1.2 is not true for solvable groups.For example,nilpotent groups,minimal non-nilpotent groups,etc.We conclude that,Example 2.1.1 Let p>1 be a prime,and suppose that m is a divisor of p-1 with|π(m)|≥2.Then there exists a group of order pm such that G is non-nilpotent,not minimal non-nilpotent,and |n(G)|≤|π(G)|-1.Theorem 2.2.1 Let G be a finite non-nilpotent group.Assume that p∈π(G)is the smallest prime such that P∈Sylp(G)is not normal in G,then |m(G)|≥|π(G)|+p.The third chapter focuses on finite p-group in which all non-maximal abelian subgroups are TI-subgroups,and give a complete classification for the minimal non-abelian p-group,the group of order p3 and of order 24 satisfying the property:Theorem 3.1 Let G be a finite minimal non-abelian p-group.Then all non-maximal abelian subgroups of G are TI-subgroups if and only if G is one of the following groups:(1)Q8;(2)Mp(n,1):=<a,b|aps=bp=1,b-1ab=a1+pn-1>,n≥2;(3)Mp(1,1,1)=<a,b,c|ap=bp=cp=1,[a,b]=c,[a,c]=[b,c]=1>.Theorem 3.2 Let G be a group of order p3.Then all non-maximal abelian subgroups of G are TI-subgroups.Theorem 3.3 Let G be a group of order 24.Then all non-maximal abelian subgroups of G are TI-subgroups if and only if G is one of the following groups:(A)Abelian groups.(B)Four non-commutative groups with cyclic maximal subgroups:(1)Generalized quaternion group:G=(a,b|a8=1,b2=a4,b-1ab=a-1);(2)Dihedral group:G=<a,b|a8=1,b2=1,b-1ab=a-1);(3)Semidihedral group:G=<a,b|a8=1,b2=1,b-1ab=a3>;(4)General metacyclic group:G=<a,b| a8=1,b2=1,b-1ab=a5>.(C)Two non-commutative groups without elements of order 8:(5)G≌Q8×G2;(6)G=<a,b,c|a4=b2=c2=1,[c,b]=a2,[a,b]=[a,c]=1>≌D8×C4.