In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup.
I knew that the set of Torsion elements is closed under the inverse operation, but was later told that it is not closed under multiplication, therefore disproving the claim. However, I couldn't find any examples of a group $G$ and two elements $a,b$ such that both $a$ and $b$ have a finite order, but $ab$ doesn't. I know that for this to happen $G$ must be an infinite and non-Abelian group, but still couldn't find a valid example.
What are some examples of groups/elements fulfilling the aforementioned property?
the group of isometries of the real line is generated by translations and reflections, with, for $a \in \mathbb{R}$: $$ T_a: x \to x + a \\ R_a: x \to 2a - x $$
So $R_a$ and $R_b$ are both involutions (elements of order 2), but their product: $R_b \circ R_a: x \to 2b - (2a -x) = x + 2(b - a) = T_{2(b-a)}(x)$. If $a \ne b$ the translation $T_{2(b-a)}(x)$ has infinite order.