Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 8 / i / Solution

Interpret the functions as maps on $\mathbb{R}\setminus \{0\}$, so every reciprocal is defined. Let $e(x)=x$, $a(x)=-x$, $b(x)=1/x$ and $c(x)=-1/x$. With the left factor applied last, direct function composition gives the Cayley tableThe table proves closure and identifies $e$ as the identity. Every element is its own inverse. Associativity is inherited from composition of maps, so the set is a group. The three nonidentity elements are commuting involutions.

Its subgroups areThere are no others: a subgroup containing two distinct nonidentity elements also contains their product, the third, and hence is the whole group; a subgroup containing just one has order two.

The abstract dihedral group $D_2$ has commuting generators $R,m$ with $R^2=m^2=I$, for example a rectangle's half-turn and reflection in one axis. The assignment $I\mapsto e$, $R\mapsto a$, $m\mapsto b$, $Rm\mapsto c$ respects products and is bijective. ThusThis is the group generated by sign reversal and reciprocation, a realization of the Klein four-group. The nonzero domain is essential; treating these as functions defined at zero would not give the stated group.