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Abstract The starting point was to investigate the relation between the order of a finite group and its structural properties, which has been given in the topic of many classical and new researches. The number of elements of a finite algebraic structure generally tells us very little about its structure. In the case of finite groups, the order of the group does not promise much more.However, we know that groups of prime orders are commutative, and groups of prime power orders are nilpotent. |