Paper 3, Section II, D

(a) Let $G$ be the dihedral group of order $4 n$, the symmetry group of a regular polygon with $2 n$ sides.

Determine all elements of order 2 in $G$. For each element of order 2 , determine its conjugacy class and the smallest normal subgroup containing it.

(b) Let $G$ be a finite group.

(i) Prove that if $H$ and $K$ are subgroups of $G$, then $K \cup H$ is a subgroup if and only if $H \subseteq K$ or $K \subseteq H$.

(ii) Let $H$ be a proper subgroup of $G$, and write $G \backslash H$ for the elements of $G$ not in $H$. Let $K$ be the subgroup of $G$ generated by $G \backslash H$.

Show that $K=G$.

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