(i) State and prove Maschke's theorem for finite-dimensional representations of finite groups.

(ii) $S_{3}$ is the group of bijections on $\{1,2,3\}$. Find the irreducible representations of $S_{3}$, state their dimensions and give their character table.

Let $T_{2}$ be the set of objects $T_{2}=\left\{a_{i_{1} i_{2}}: i_{1}, i_{2}=1,2,3\right\}$. The operation of the permutation group $S_{3}$ on $T_{2}$ is defined by the operation of the elements of $S_{3}$ separately on each index $i_{1}$ and $i_{2}$. For example,

$$P_{12}: a_{13} \rightarrow a_{23}, \quad P_{231}: a_{23} \rightarrow a_{31}, \quad P_{13}: a_{33} \rightarrow a_{11}$$

By considering a representative operator from each conjugacy class of $S_{3}$, find the table of group characters for the representation $\mathcal{T}_{2}$ of $S_{3}$ acting on $T_{2}$. Hence, deduce the irreducible representations into which $\mathcal{T}_{2}$ decomposes.