Abstract:  Since the hypergroup was raised by prof.Li Hongxing and Wang Peizhuang in 1985,the problem of structure of hypergroup on the cyclical group has been unsolved.This paper solved the structure of hypergrup on the cyclical group completely.In the main,proved the two structure theorems of the associative hypergroup for finite and infinite cyclical groups,and intruduced the forming of examples of hypergroup on cyclical group.

Key words: upgrade, hypergroup, trivil hypergroupicyclical group