Research On T. Asai & T. Yoshida Conjecture Of Group Homomorphism

The conjecture on the relationship between the number of group homomorphisms and group order proposed by Japanese scholars T.Asai and T.Yoshida has not been solved up to now.This dissertation explores this issue on the basis of previous studies.Firstly,we establish a one-to-one correspondence between homomorphism set from an infinite cyclic group to a finite group and this finite group.Secondly,we discuss that the direct product decomposition of a group influence on the group homomorphism number of direct integral decomposition.Through decomposition theorem,we prove that the homomorphism number from a free abelian group with rank r to a finite group is the multiple of the order of G.At the same time,the result of decomposition theorem generalizes a well-known result among group homomorphisms.Finally,using the influence of group direct product decomposition on the number of group homomorphisms and the minimal counter-example method,we discuss T.Asai and T.Yoshida conjectures hold when two groups meet certain conditions.