High Quality Content by WIKIPEDIA articles! In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element. Such groups are also called p-primary or simply primary. A finite group is a p-group if and only if its order (the number of its elements) is a power of p. The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prufer group, and for an example of an infinite simple p-group, see Tarski monster group.

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