If G is a finite group generated by $a,b \in G$, both $a$ and $b$ having order 2, what can we say about the order of another element $xy \in G$?
I was thinking that we can conclude that $xy$ has finite order because it is contained in G, but nothing else can be concluded. Is this correct or am I missing something?
Consider the dihedral group $D_n$, a rotation may be written as a product of two reflections (each of which has order $2$), thus if we choose a suitable $n$, we can show that the product of two elements of order $2$ can have any order. (Of course, if we require that it is an element in a finite group, its order is finite).