if we let $k=\mathbb{C}$, I need to show that the group algebra $kS_2$ isomorphic to direct product of matrix algebras. Another question is if it is true for an arbitrary field $k$. If we let $S_2=\{e,s\}$ I was told to look at $b_1=(e+s)/2,b_2=(e-s)/2$.
An element $x\in\mathbb{C}S_2$ looks like $x=c_1e+c_2s,c_1,c_2\in\mathbb{C}$ or equivalently, a function with value $c_1,c_2$ at $e,s$ respectively. I also know that $\mathbb{C}$ is isomorphic to the group algebra generated by $\begin{pmatrix}x &y\\-y&x\end{pmatrix},x,y\in\mathbb{R}$. I failed to find a product of matrix algebras that is suitable here. I need tips but I don't want a solution straight away.
Hints:
By Maschke's theorem, the algebra group $\;\Bbb CG\;$ , for any finite group $\;G\;$ , is semisimple.
Now just apply Artin-Wedderburn's Theorem