MHB Show that there are y,z such that y,z commute and their order is m and n

[mathmari]

Hey!

I got stuck at the following exercise:

If $x \in G$ has order $mn$ with $ (m,n)=1 $, show that there are $y,z$ with $ x=yz $ such that $y$,$z$ commute and they have order $m$ and $n$ respectively.

Could you give me some hints?? (Wondering)

 

[Opalg]

Hey!

I got stuck at the following exercise:

If $x \in G$ has order $mn$ with $ (m,n)=1 $, show that there are $y,z$ with $ x=yz $ such that $y$,$z$ commute and they have order $m$ and $n$ respectively.

Could you give me some hints?? (Wondering)

Hint: think about powers of $x$.

 

[Euge]

Another hint: Since $(m,n) = 1$, there are integers $s$ and $t$ such that $1 = sm + tn$.