Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?

Question

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice)

Let x be an element of order r of a group G, and let y be an element of G' of order s. What is the order of $(x, y)$ in the product group G x G'?

The answer is $lcm(r,s)$, as answered here in this question $x$ in group G with order $r$, $y$ in group $G'$ with order $s$ what is the order of $(x,y)$ in $G$ x $G'$.

Artin's Algebra also has a proposition

Proposition 2.11.3 Let r and s be relatively prime integers. A cyclic group of order rs is isomorphic to the product of a cyclic group of order r and a cyclic group of order s.

What is the relationship between these and Order of Product of Disjoint Permutations, which says that a product of disjoint permutations $\pi = \rho_1 \rho_2 ... \rho_r$ with orders $k_1,k_2,...,k_r$ has order $lcm\{k_1,k_2,...k_r\}$?

Answer 1

The property concerning the order of a product of disjoint permutations comes from a more general fact:

Let $G$ be a group, $a\in G$ of order $r$ and $b\in G$ of order $s$.

If $ab=ba$, then the order of $ab$ is $\operatorname{lcm}(r,s)$.

Since disjoint permutations commute, the result follows.

Finally, how does this relate to the situation of a product of groups? Well, let $G,G'$ be groups, $a\in G$ of order $r$ and $b\in G'$ of order $s$. Then $(a,e)\in G\times G'$ has order $r$ and $(e,b)\in G\times G'$ has order $s$. Also, notice that $(a,e)\cdot(e,b)=(a,b)=(e,b)\cdot(a,e)$. Hence, the result on the order of $(a,b)$ follows from the fact above.