P-group

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 g^m 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}.