Study On The Number Of Homomorphisms Between Two Classes Of Non-commutative Groups And Quaternion Groups

In this dissertation,study the basic structure of a class of internal commutative p-groups M_p（n,m,1）,ternary generating groups G₁ and quaternion groups Q_4r.Through the structural features of the generators and the generating relations of groups,taking a class of internal commutative p-groups M_p（n,m,1）=<a,b,c|a^pn=b^pm=c^p=1,[a,b]=c,[c,a]=[c,b]=1 and a class of the ternary generating groups G₁=<d,f,g|d^u=f²=g²=1,d^f=d^-1,f^g=f^-1,g^d=g^-1>as the research object,and the homomorphic images between these two classes of noncommutative groups and the quaternion group Q_4r are constructed,and the number of homomorphisms between the quaternion groups Q_4r and the internal commutative p-groups M_p（n,m,1）and the ternary generating groups G₁ constructed by the semidirect product cyclic group of the dihedral groups are calculated.The conjecture of T.Asai and T.Yoshida is verified,and expectations are made for follow-up research.