Paper 3, Section II, D

(a) Let $G$ be a finite group and let $H$ be a subgroup of $G$. Show that if $|G|=2|H|$ then $H$ is normal in $G$.

Show that the dihedral group $D_{2 n}$ of order $2 n$ has a normal subgroup different from both $D_{2 n}$ and $\{e\}$.

For each integer $k \geqslant 3$, give an example of a finite group $G$, and a subgroup $H$, such that $|G|=k|H|$ and $H$ is not normal in $G$.

(b) Show that $A_{5}$ is a simple group.

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