order of elements in finite groups

This article proves two elementary results regarding the orders of group elements in finite groupsMathworldPlanetmath.

Theorem 1

Let G be a finite group, and let aG and bG be elements of G that commute with each other. Let m=|a|, n=|b|. If gcd(m,n)=1, then mn=|ab|.

Proof. Note first that


since a and b commute with each other. Thus |ab|mn. Now suppose |ab|=k. Then


and thus n|km. But gcd(m,n)=1, so n|k. Similarly, m|k and thus mn|k=|ab|. These two results together imply that mn=k.

Theorem 2

Let G be a finite abelian group. If G contains elements of orders m and n, then it contains an element of order lcm(m,n).

Proof. Choose a and b of orders m and n respectively, and write


where the pi are distinct primes. Thus for each i, either pikim or pikin. Thus either am/piki or bn/piki has order piki. Let this element be ci. Now, the orders of the ci are pairwise coprime by construction, so


and thus ci is the required element.

Title order of elements in finite groups
Canonical name OrderOfElementsInFiniteGroups
Date of creation 2013-03-22 16:34:02
Last modified on 2013-03-22 16:34:02
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 20A05