Paper 3, Section II, $7 \mathrm{D}$

State and prove the orbit-stabiliser theorem.

Let $p$ be a prime number, and $G$ be a finite group of order $p^{n}$ with $n \geqslant 1$. If $N$ is a non-trivial normal subgroup of $G$, show that $N \cap Z(G)$ contains a non-trivial element.

If $H$ is a proper subgroup of $G$, show that there is a $g \in G \backslash H$ such that $g^{-1} H g=H$.

[You may use Lagrange's theorem, provided you state it clearly.]

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