Equivalence of definitions of group of prime power order

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) uses::Order of element divides order of group
 * 2) uses::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 $$q$$.

(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).