关键词: 上循环, 群作用, 群同态, 有限群

Abstract: By using the equivalence class of group action,  the set of cocycles and the group homomorphism were connected. By characterizing the number of homomorphisms between two finite groups through the set of cocycles,  we proved that for any finite groups A,G, if the number of elements in the set of cocycles of  A and G could be  divided by the greatest common divisor of |A| and |G|, the number of homomorphisms between A and G could be divided by the greatest common divisor of |A| and |G|.

Key words: cocycle, group action, homomorphism, finite group