(a) Let $G$ be a finite group acting on a finite set $X$. For any subset $T$ of $G$, we define the fixed point set as $X^{T}=\{x \in X: \forall g \in T, g \cdot x=x\}$. Write $X^{g}$ for $X^{\{g\}}(g \in G)$. Let $G \backslash X$ be the set of $G$-orbits in $X$. In what follows you may assume the orbit-stabiliser theorem.

Prove that

$$|X|=\left|X^{G}\right|+\sum_{x}|G| /\left|G_{x}\right|$$

where the sum is taken over a set of representatives for the orbits containing more than one element.

By considering the set $Z=\{(g, x) \in G \times X: g \cdot x=x\}$, or otherwise, show also that

$$|G \backslash X|=\frac{1}{|G|} \sum_{g \in G}\left|X^{g}\right|$$

(b) Let $V$ be the set of vertices of a regular pentagon and let the dihedral group $D_{10}$ act on $V$. Consider the set $X_{n}$ of functions $F: V \rightarrow \mathbb{Z}_{n}$ (the integers mod $\left.n\right)$. Assume that $D_{10}$ and its rotation subgroup $C_{5}$ act on $X_{n}$ by the rule

$$(g \cdot F)(v)=F\left(g^{-1} \cdot v\right),$$

where $g \in D_{10}, F \in X_{n}$ and $v \in V$. It is given that $\left|X_{n}\right|=n^{5}$. We define a necklace to be a $C_{5}$-orbit in $X_{n}$ and a bracelet to be a $D_{10}$-orbit in $X_{n}$.

Find the number of necklaces and bracelets for any $n$.