Find and example of two elements $a,b$ in a finite group $G$ such that $|a| = |b| = 2, a \ne b$ and $|ab|$ is odd. Find and example of two elements $a,b$ in a finite group $G$ such that $|a| = |b| = 2, a \ne b$ and $|ab|$ is odd. Any ideas as to how I would go about finding it?
Thanks

There is a very general example you should know about, that of dihedral groups. A dihedral group has order $2n$, for any $n \ge 2$, and it is generated by two elements of order $2$, whose product has order $n$.
Probably the simplest way to see these groups is as a group of bijective maps on $\mathbf{Z}_{n}$,
$$
a : x \mapsto -x, \qquad b: x \mapsto -x-1.
$$
(Coefficients are written as integers, but meant to be in $\mathbf{Z}_{n}$.) These are clearly elements of order $2$, while because of
$$
a \circ b(x) = (a(b(x)) = -(-x - 1) = x + 1 
$$
$ab = a \circ b$ has order $n$.
(Geometrically, such a group is the group of congruences of a regular $n$-gon.)
If you do the same thing over $\mathbf{Z}$, you find that $ab$ has infinite order.
