nLab prime power

Definition

In the natural numbers, a prime power $p^n$ is a positive power $n \geq 1$ of a prime number $p$.

 Properties

The fundamental theorem of arithmetic states that every positive natural number is a finite product of prime powers.

Uses in other parts of mathematics

In group theory, a group is a p-primary group if its order is a prime power of the given prime number $p$.

The fundamental theorem of finite abelian groups states that every finite abelian group is the direct sum of cyclic groups of prime power order.

In ring theory, the integers modulo n which are local rings are precisely the integers modulo a prime power.

See also

• prime number

External links

• Wikipedia, Prime power.