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

Statement

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

1. The order of the group is a power of $p$.
2. The order of every element in the group is a power of $p$.

Facts used

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

Proof

(1) implies (2)

This follows directly from fact (1).

(2) implies (1)

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