For any integers $m,n,r>1$ , does there exist an infinite group $G$,with elements $a,b\in G$ such that $o(a)=m , o(b)=n , o(ab)=r $?

Question

I know that for any integers $m,n,r$ , all greater than $1$ , there is a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n , o(ab)=r $ ; now in the proof , that I know , the group $G$ we construct is $SL_2(F)/\{I,-I\}$ where $F$ is a finite field depending on $m,n,r$ ; so the group $G$ is finite ; so I would like to ask

for any integers $m,n,r$ all greater than $1$ , does there exist an infinite group $G$ (depending on $m,n,r$ may be ) with elements $a,b \in G$ such that $o(a)=m , o(b)=n , o(ab)=r $ ?

Answer 1

You can take your finite group $G$ and add $\mathbb Z$ like this: $G\times\mathbb Z$, so it becomes infinite, but still has the desired property as it contains an isomorphic copy of $G$.