Group generated by sign reversal and reciprocation ($a(x)=-x,b(x)=1/x,a^2=b^2=e,ab=ba$)

On the nonzero real numbers, sign reversal $a$ and reciprocation $b$ are commuting involutions. Their function composition generates exactly $e(x)=x$, $a(x)=-x$, $b(x)=1/x$ and $ab(x)=-1/x$. The relations $a^2=b^2=e$ and $ab=ba$ identify the group with $C_2\times C_2$, the Klein four-group. Its subgroups are the trivial group, its three order-two cyclic subgroups and itself. Excluding zero from the domain is necessary for all four transformations to be defined and invertible.