p-group

A p-group is a group built entirely out of a single prime: every element has order some power of p. For finite groups this is the same as saying the group's order is a power of p. These are the “atoms of prime power order,” and the Sylow theorems make them the building blocks from which the prime-by-prime structure of any finite group is assembled.

Formally, for a fixed prime p, a group is a p-group if every element has order p^n for some n ≥ 0 (depending on the element). A finite group is a p-group exactly when |G| = p^k for some k, by Cauchy's theorem and Lagrange. Their defining structural feature, proved from the class equation, is that a nontrivial finite p-group always has a nontrivial center.

From the nontrivial-center fact much follows: finite p-groups are nilpotent, they possess a normal subgroup of every order dividing |G| (a full chain of normal subgroups with quotients of order p), and they are never simple unless of order p. The infinite theory is richer and stranger — the Prüfer p-group Z(p^∞) is an infinite p-group every proper subgroup of which is finite — so the clean finite picture should not be over-extended.