The order of any element can be divisible by the order of a group. Now the order of a group is prime number P, so the order of an element is either 1 or P. There is only one unit element in G, and the order of other elements is not equal to 1, so they are all P. Take any non-unit element whose order is equal to P, then the order of the cyclic subgroup of G generated by it is also P, which is equal to the whole group G, so G is equal to the cyclic group generated by any non-unit element. Complete the certificate.

The finite group g is a p group.

If and only if the order of g is the power of p. The study of finite P groups is an important topic in finite group theory. Shiloh theorem shows that the structure of finite P subgroups has great influence on the properties of the whole group. The study of finite nilpotent groups can be summed up as the study of finite P groups. A series of work on P- groups by Hall (P.) from 1930 to 1950 had a profound influence on the research of P- groups. Classifying finite P groups according to isomorphism is the basic task of finite P group research, but it is an extremely difficult task, which is far from being solved at present.