Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 9 / ii / c / Solution

The conjugacy class of $g$ is $\{hg{h}^{-1}:h\in G\}$. Rotations commute with one another, while conjugation by a reflection changes $R^j$ to $R^{-j}$. Thus the rotation classes are $\{I\}$, $\{R,R^4\}$ and $\{R^2,R^3\}$.

For reflections,Because multiplication by two permutes the residues modulo five, these conjugates exhaust all five reflections. Consequently the complete class partition isTheir sizes $1+2+2+5=10$ account for every element, as in the conjugacy classes of a finite dihedral group.

A normal subgroup is a union of complete conjugacy classes. The trivial subgroup is normal, and the rotation subgroup is normal because conjugation preserves it; equivalently it has index two. None of the five order-two subgroups is normal, because it contains just one member of the five-element reflection class. Thus the proper normal subgroups are exactly