order of elements in finite groups
This article proves two elementary results regarding the orders of group elements in finite groups.
Let be a finite group, and let and be elements of that commute with each other. Let , . If , then .
Proof. Note first that
since and commute with each other. Thus . Now suppose . Then
and thus . But , so . Similarly, and thus . These two results together imply that .
Let be a finite abelian group. If contains elements of orders and , then it contains an element of order .
Proof. Choose and of orders and respectively, and write
where the are distinct primes. Thus for each , either or . Thus either or has order . Let this element be . Now, the orders of the are pairwise coprime by construction, so
and thus is the required element.