Representation theory of finite group $G$ and $K[G]$-module Let $G$ a finite group and $K$ a field. We define the set 
$ K[G]:=\lbrace f: x \in G \longrightarrow \alpha_x \in K \rbrace$
$f$ is an application and $\alpha \in K$. With usual operations, $(K[G],+,\cdot)$ is a vector space, in particular it proves that $G$ is a base of $K[G]$, and it's true the notation $f(x)=\sum_{x \in G} \alpha_x x$  with $\alpha_x:=f(x)$. Now a module on $K[G]$ is by definition a right module $V$ on $K[G]$, i.e. there is a homomorphism
$\psi : K[G] \longrightarrow End_K V$ : $\psi(1_G)=1_K$
where $End_K V$ is the set of endomorphisms of $V$, $\psi$ is a homomorphism of rings, also between vector space. We say that a rapresentation of a finite group $G$ on $V$ is each homomorphism 
$\varphi : g \in G \longrightarrow \varphi(g) \in GL(V)$
where $GL(V)$ is the set of the automorphisms of $V$, use this notation: 
$v \varphi(g):=\varphi(g)(v)$  for all $v \in V$ and $g \in G$.
My problem is to show that if there is a representation of $G$ to $V$, then on $V$ is defined a structure of $K[G]-module$.
$Proof$
With the previous notations we put 
$(*)$ $$v ( \sum_{g \in G} \alpha_g g  ) := \sum_{g \in G} \alpha_g (v \varphi (g))$$
for all $v \in V$ and $\sum_{g\in G} \alpha_g g \in K[G]$, then this condition defines an homomorphism 
$$\psi : K[G] \longrightarrow GL(V) \subset End_K V$$
in fact if $f=\sum_{x \in G} \alpha_x x$ and $g=\sum_{y \in G} \beta_y y \in K[G]$, then
$$\psi(fg)=\sum_{x,y \in G} \alpha_{x}\beta_{y}\varphi(xy)=\sum_{x,y \in G} \alpha_{x}\beta_{y} \varphi(x) \varphi(y)=\sum_{x \in G} \alpha_{x}  \varphi(x) \cdot \sum_{y \in G} \beta_{y} \varphi(y)=\psi(f)\psi(g)$$
Is it correct? thank you for reply
 A: I think you have some confusion about $K[G]$. On the one hand, you have $K[G]$ as the set of functions $f:G\to K$. If that is what you mean, then multiplication is defined by convolution:
$$
(f_1f_2)(x)=\sum_{g\in G}f_1(g)f_2(g^{-1}x)
$$
If $g\in G$ and $g^*$ is the the map $g^*(h)=\delta_{g,h}$, then every element of $K[G]$ can be written as $\sum_g\alpha_g g^*$. Moreover, it is straightforward to check that $g^*h^*=(gh)^*$, so $K[G]$ is isomorphic to the algebra $KG$ of linear combinations of elements of $G$,
$$\sum_g\alpha_gg$$
with multiplication extended linearly from the multiplication in $G$, which seems to be your latter description. Hopefully you understand this identification. 
Given a representation $\rho:G\to GL(V)$, define $$\tilde{\rho}:KG\to End(V)$$
by extending $\rho$ linearly:
$$\tilde{\rho}\left(\sum_g\alpha_gg\right)=\sum_g\alpha_g\rho(g).$$
Then
\begin{align}\tilde{\rho}\left(\sum_g\alpha_gg\sum_h\beta_hh\right)&=
\tilde{\rho}\left(\sum_{g,h}\alpha_g\beta_hgh\right)\\
&=\sum_{g,h}\alpha_g\beta_h\rho(gh)\\
&=\sum_{g,h}\alpha_g\beta_h\rho(g)\rho(h)\\
&=\sum_g\alpha_g\rho(g)\sum_h\beta_h\rho(h)
\end{align}
