The Number Of Homomorphisms Between Two Special 2-groups And Some Non-Abelian Finite Groups

As one of the basic quantitative information of finite group theory,the number of homomorphism among finite groups is often used to explore the structure and properties of finite groups.In this dissertation,consider a class of inner abelian subcyclic 2-groupsM2(2,m)=<a,b|a4=b2m=1,b-1a=a-1>(m>2)of two-elements generation as the research object,constructed all homomorphism between it and generalized quaternion group,the number of endomorphism are calculated.Further,a class of ternary generated finite 2-group G obtained by direct product expansion of the group M2(2,m)is analyzed,and the number of homomorphisms between this group and the generalized quaternion group and the subcyclic group Hn,2p and its endomorphisms are calculated.As an application,it is verified that these groups satisfy the conjecture of T.Asai and T.Yoshida.The dissertation is divided into five chapters.Chapter one,it introduces the related definitions and lemmas used in this dissertation.Chapter two,the number of homomorphisms between the inner abelian subcyclic 2-groups M2(2,m)and the generalized quaternion group Q4n is calculated.Chapter three,the number of homomorphisms between a class of finite 2-groups G and some noncommutative groups.Chapter four,the number of homomorphisms between a class of finite 2-groups G is calculated.Chapter fiver,it is proved that all above groups satisfy the conjecture of T.Asai and T.Yoshida based on the calculation results of the previous four Chapter.