The Number Of Homomorphisms From A Class Of Non-Abelian Finite Groups With Order 10p~n Into Some Finite Groups

The study of the number of homomorphisms among finite groups is a meaningful work in the theory of finite groups,which is closely related to the isomorphism classification of finite groups.In this dissertation,we will consider the non-abelian finite groups with all cycle of Sylow p-subgroups,and select G10pn=<a,b|a10=1=bpn,ba=ab-1>(p>5 is prime)as the research object because it is a non-isomorphic form of 10pn order non-abelian group with all cycle of Sylow p-subgroups whose isomorphism classification is clear.With the combination of the structures and properties of the group G10pn,we will build the homomorphism mapping between the group G10pn and the quaternion group Q4m,the group G10pn and the module group Mqm,and the endomorphism of the group G10pn,where the quaternion group Q4m and the module group Mqm are all non-abelian groups of two-elements generation.Further,the number of homomorphism and endomorphism is calculated respectively.As an application,we will verify that these groups satisfy the conjecture of Asai and Yoshida by the properties of commutator groups of related groups and the number of homomorphisms between these groups.The dissertation is divided into five chapters.Chapter one,it introduces the definitions and related lemmas used in this paper.Chapter two,the number of homomorphisms between the group G10pn and the quaternion group Q4m is calculated.Chapter three,the number of homomorphisms between group G10pn and modular group Mqm is calculated.Chapter four,the homomorphism of between the group G10pn is built and the number of homomorphism is calculated.Chapter five,it is proved that all above three non-abelian groups of two-elements generation satisfy the conjectures of Asai and Yoshida based on the calculation results of the previous four Chapters.