= Group generated by sign reversal and reciprocation
{title2=$a(x)=-x,\quad b(x)=1/x,\quad a^2=b^2=e,\quad ab=ba$}
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.
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