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

(a) State the orbit-stabilizer theorem.

Let a group $G$ act on itself by conjugation. Define the centre $Z(G)$ of $G$, and show that $Z(G)$ consists of the orbits of size 1 . Show that $Z(G)$ is a normal subgroup of $G$.

(b) Now let $|G|=p^{n}$, where $p$ is a prime and $n \geqslant 1$. Show that if $G$ acts on a set $X$, and $Y$ is an orbit of this action, then either $|Y|=1$ or $p$ divides $|Y|$.

Show that $|Z(G)|>1$.

By considering the set of elements of $G$ that commute with a fixed element $x$ not in $Z(G)$, show that $Z(G)$ cannot have order $p^{n-1}$.

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