Let $G$ be a group and let $A$ be a non-empty subset of $G$. Show that

$$C(A)=\{g \in G: g h=h g \quad \text { for all } h \in A\}$$

is a subgroup of $G$.

Show that $\rho: G \times G \rightarrow G$ given by

$$\rho(g, h)=g h g^{-1}$$

defines an action of $G$ on itself.

Suppose $G$ is finite, let $O_{1}, \ldots, O_{n}$ be the orbits of the action $\rho$ and let $h_{i} \in O_{i}$ for $i=1, \ldots, n$. Using the Orbit-Stabilizer Theorem, or otherwise, show that

$$|G|=|C(G)|+\sum_{i}|G| /\left|C\left(\left\{h_{i}\right\}\right)\right|$$

where the sum runs over all values of $i$ such that $\left|O_{i}\right|>1$.

Let $G$ be a finite group of order $p^{r}$, where $p$ is a prime and $r$ is a positive integer. Show that $C(G)$ contains more than one element.