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Given a prime number p, a p-group P is a mathematical group all elements have a power of p as their order (that is, for each g in P, there exists an integer n such that g to the power pn is equal to 1, while gm is not 1 for any m<pn). If G is finite, this implies that the order of G (the number of its elements) is itself a power of p.

Quite a lot is known about the structure of finite p-groups. One of the first standard results is that the center of a finite p-groups cannot be the trivial subgroup {1}.