Order of the Product of Two Elements in an Abelian Group

Abelian Group problems and solutions

Problem 205

Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively.
If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$.

 
FavoriteLoadingAdd to solve later

Sponsored Links

Proof.

Let $r$ be the order of the element $ab$.
Since we have
\begin{align*}
(ab)^{mn}&=a^{mn}b^{mn} \quad \text{ (since } G \text{ is an abelian group)}\\
&=(a^m)^n(b^n)^m\\
&=1
\end{align*}

since $a^m=1$ and $b^n=1$.
This implies that the order $r$ of $ab$ divides $mn$, that is, we have
\[ r |mn. \tag{*}\]

Now, since $r$ is the order of $ab$ we have
\[1=(ab)^r=a^rb^r.\] Then we have
\begin{align*}
1=1^n=a^{rn}b^{rn}=a^{rn}
\end{align*}

since $b^n=1$. This yields that the order $m$ of the element $a$ divides $rn$.

Since $m$ and $n$ are relatively prime, this implies that we have
\[m|r.\]

Similarly (switch the role of $n$ and $m$), we obtain
\[n|r.\] Thus we have
\[mn|r \tag{**}\] since $m$ and $n$ are relatively prime.

From (*) and (**), we have $r=mn$, and hence the order of the element $ab$ is $mn$.

Related Question.

As a generalization of this problem, try the following problem.

Problem.Let $G$ be an abelian group.
Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.
Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.

A proof of this problem is given in the post “The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements“.

Also See the post “Order of product of two elements in a group” for a similar problem about the order of elements in a non-abelian group.


FavoriteLoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

2 Responses

  1. 03/28/2017

    […] the post “Order of the product of two elements in an abelian group” for a similar problem about the order of elements in abelian […]

  2. 06/30/2017

    […] Recall that if the orders $m, n$ of elements $a, b$ of an abelian group are relatively prime, then the order of the product $ab$ is $mn$. (For a proof, see the post “Order of the Product of Two Elements in an Abelian Group“.) […]

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Group Theory
Group Theory Problems and Solutions in Mathematics
Two Normal Subgroups Intersecting Trivially Commute Each Other

Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$....

Close