The Number Of Cyclic Subgroups And Structure Of Groups

The cyclic subgroups are fundamental and very important subgroups of groups.In this thesis we study the number of cyclic subgroups and structure of groups.Let G be a finite group and I(G)denote the number of elements g?G with g2=1 in G.Suppose that c(G)is the number of cyclic subgroups of G and?(G)the function c(G)/|G|.This thesis consists of four chapters.In the first chapter,we introduce the basic definitions and symbols,and the research background and research progress on the number of cyclic subgroup.In the second chapter,we discuss the groups with ?(G)=3/4.According to the Garonzi and Lima's classification of groups with ?(G)>3/4 and Miller and Nekrasov's with I(G)=1/2|G|,we classify these completely.It proved that if ?(G)=3/4,then G are isomorphic to a direct product of an elementary abelian 2-group and a dihedral group D16,D24,or a group satisfied exp(G)=4 and I(G)=1/2|G|.In the third chapter,we discuss some groups which are satisfied with 17/24??(G)<3/4 and the maximal primer factor of |G| is not less than 2 and such groups are direct product of an elementary abelian 2-group and a group X isomorphic to one of the following groups:SmallGroup(24,8),SmallGroup(72,33),SmallGroup(96,81),S4,S3,D8.In the last chapter,we classify the structure of finite solvable groups whose maximum prime divisor is not less than 17 and ?(G)>8/15.It proved that those groups are isomorphic to a direct product of an elementary abelian 2-group and one of the following groups:D102,D114,D136,D152,(Z68×Z4)(?)Z2 or D2p,where p?{17,19,23,29}.