The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements

Problem 497

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.

 
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Hint.

First, consider the case when $m$ and $n$ are relatively prime.

Proof.

When $m$ and $n$ are relatively prime

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“.)

So if $m, n$ are relatively prime, then we can take $c=ab\in G$ and $c$ has order $mn$, which is the least common multiple.

The general Case

Now we consider the general case.

Let $p_i$ be the prime factors of either $m$ or $n$.
Then write prime factorizations of $m$ and $n$ as
\[m=\prod_{i}p_i^{\alpha_i} \text{ and } n=\prod_{i} p_i^{\beta_i}.\] Here $\alpha_i$ and $\beta_i$ are nonzero integers (could be zero).
Define
\[m’=\prod_{i: \alpha_i \geq \beta_i}p_i^{\alpha_i} \text{ and } n’=\prod_{i: \beta_i> \alpha_i} p_i^{\beta_i}.\]

(For example, if $m=2^3\cdot 3^2\cdot 5$ and $n=3^2\cdot 7$ then $m’=2^3\cdot 3^2\cdot 5$ and $n’=7$.)

Note that $m’\mid m$ and $n’\mid n$, and also $m’$ and $n’$ are relatively prime. The least common multiple $l$ of $m$ and $n$ is given by
\[l=m’n’\]

Consider the element $a’:=a^{m/m’}$. We claim that the order of $a’$ is $m’$.

Let $k$ be the order of the element $a’$. Then we have
\begin{align*}
e=(a’)^k=(a^{\frac{m}{m’}})^k=a^{mk/m’},
\end{align*}
where $e$ is the identity element in the group $G$.
This yields that $m$ divides $mk/m’$ since $m$ is the order of $a$.
It follows that $m’$ divides $k$.

On the other hand, we have
\begin{align*}
(a^{m/m’})^{m’}=a^m=e,
\end{align*}
and hence $k$ divides $m’$ since $k$ is the order of the element $a^{m/m’}$.

As a result, we have $k=m’$.
So the order of $a’$ is $m’$.

Similarly, the order of $b’:=b^{n/n’}$ is $n’$.

The orders of elements $a’$ and $b$ are $m’$ and $n’$, and they are relatively prime.
Hence we can apply the first case and we conclude that the element $a’b’$ has order
\[m’n’=l.\] Thus, we can take $c=a’b’$.

The Case When $G$ is a Non-Abelian Group

Next, we show that if $G$ is a non-abelian group then the statement does not hold.

For example, consider the symmetric group $S_3$ with three letters.
Let
\[a=(1\,2\,3) \text{ and } b=(1 \,2).\]

Then the order of $a$ is $3$ and the order of $b$ is $2$.
The least common multiple of $2$ and $3$ is $6$.
However, the symmetric group $S_3$ have no elements of order $6$.

Hence the statement of the problem does not hold for non-abelian groups.

Related Question.

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

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