# Order of Product of Two Elements in a Group ## Problem 354

Let $G$ be a group. Let $a$ and $b$ be elements of $G$.
If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample. Add to solve later

## Proof.

We claim that it is not true. As a counterexample, consider $G=S_3$, the symmetric group of three letters.
Let $a=(1\, 2), b=(1 \,3)$ be transposition elements in $S_3$.
The orders of $a$ and $b$ are both $2$.

Consider the product
$ab=(1\, 2)(1 \,3)=(1 \, 3 \, 2).$ Then it is straightforward to check that the order of $ab$ is $3$, which does not divide $4$ (the product of orders of $a$ and $b$).

Therefore, the group $G=S_3$ and elements $a=(1\, 2), b=(1 \,3)\in G$ serve as a counterexample.

## Remark. (Abelian group case)

If we further assume that $G$ is an abelian group, then the statement is true.
Here is the proof if $G$ is abelian.

Let $e$ be the identity element of $G$.
\begin{align*}
(ab)^{mn} &=a^{mn}b^{mn} && \text{ since $G$ is abelian}\\
&=(a^m)^n(b^n)^m\\
&=e^n e^m && \text{since the order of $a, b$ are $m, n$ respectively}\\
&=e.
\end{align*}
Thus the order of $ab$ divides $mn$.

## Related Question.

If the group is abelian, then the statement is true.

Problem. 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 $mn$.

See the post “Order of the Product of Two Elements in an Abelian Group” for a proof of this problem.

More generally, we can prove the following.

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 $n$.

Also determine whether the statement is true if $G$ is a non-abelian group.

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“. Add to solve later

### 2 Responses

1. 06/30/2017

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

2. 10/25/2017

[…] For a solution of this problem, see the post “Order of Product of Two Elements in a Group“. […]

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