# Prime number

From Groupprops

## Definition

A **prime number** is a natural number greater than such that it has exactly two divisors that are natural numbers: and the number itself.

The first few prime numbers are .

## Facts

- For every prime number, there is a unique group whose order is that prime number. This is the cyclic group of that order. Further, the groups of prime order are precisely the same as the simple abelian groups.
`Further information: Group of prime order, equivalence of definitions of group of prime order` - There are infinitely many prime numbers.
`Further information: Number:Infinitude of primes` - A prime power is a number that is the power of a prime -- in other words, there is at most one prime in its factorization. A group whose order is a prime power is termed a group of prime power order. Any group of prime power order is a nilpotent group.
`Further information: Prime power order implies nilpotent`