Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A4 / a / iii

With $R=C_2(z)$, $S={\sigma }_{xz}$ and $T={\sigma }_{yz}$, composition givesEvery entry belongs to the set, proving closure. $E$ is the identity element; every element is its own inverse. Associativity follows because these are actual spatial transformations: composing functions is associative. Thus all group axioms hold, and the group is isomorphic to the Klein four-group.

For a concrete matrix check, use $E=\mathrm{diag}(1,1,1)$, $R=\mathrm{diag}(-1,-1,1)$, $S=\mathrm{diag}(1,-1,1)$ and $T=\mathrm{diag}(-1,1,1)$. Their matrix products reproduce the table.