prime power in nLab

Contents

Definition

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

 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 pp.

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.