Z mod 2 power 3/Finite proper symmetry group/Exercise

a) Show that the group ${}\mathbb {Z} /(2)\times \mathbb {Z} /(2)\times \mathbb {Z} /(2)$ is not the proper symmetry group of a subset ${}T\subseteq \mathbb {R} ^{3}$.

b) Show that the group ${}\mathbb {Z} /(2)\times \mathbb {Z} /(2)\times \mathbb {Z} /(2)$ can be realized as a subgroup of the full isometry group ${}\operatorname {O} _{3}\!{\left(\mathbb {R} \right)}$.

c)

Consider the proper symmetry group of a rectangular cuboid with three different side lengths. For every axis given by the centers of two opposite faces, the half rotation around this axis is symmetry. Does this contradict part (a)?

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