# Equivalence of definitions of group of prime power order

This article gives a proof/explanation of the equivalence of multiple definitions for the term group of prime power order

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

The following are equivalent for a finite group and a prime :

- The order of the group is a power of .
- The order of every element in the group is a power of .

## Facts used

- Order of element divides order of group
- Cauchy's theorem: This states that if is a prime dividing the order of a finite group, there exists an element in the finite group of order .

## Proof

### (1) implies (2)

This follows directly from fact (1).

### (2) implies (1)

Suppose is a finite group satisfying (2). Then, by fact (2), if is any prime other than dividing the order of , has anelement of order , contradicting (2). Thus, is the only prime dividing the order of , proving (1).