The concept of ‎the ‎group algebra I have to read the concept of  ‎the ‎group algebra from the book " Matej ‎Brešar‎" ‎‎‎‎to solve the problem, but I have a problem with this ‎concept.‎
The concept of  ‎the ‎group algebra:
Assume temporarily that $G$ is an arbitrary set. If $F$ is, as always, a field, then we can form the vector space over $F$ whose basis is $G$. Its elements are formal $\sum_‎{‎g ‎\in G‎ ‎}‎ ‎\lambda‎_{g} g$ where $λ_g \in F$ and all but finitely many $\lambda‎_{g}$ are zero. The definitions of addition and
scalar multiplication are self-explanatory. Assume now that $G$  is a group. Then this vector space becomes an algebra if we define multiplication by simply extending the group multiplication on $G$ to the whole space; taking into account the algebra axioms this can obviously be done in a unique way. Thus,
‎‎
$ ‎(‎\sum_‎{‎g ‎\in G‎ ‎}‎ ‎\lambda‎_{g} g‎ ‎) ‎(‎\sum_‎{‎h ‎‎\in ‎G}‎‎ ‎‎\mu‎‎_{‎h‎} ‎h‎ ‎) ‎=‎\sum‎‎‎_{‎k \in G‎}‎ ‎\nu_k k ‎‎‎‎ \nu‎_{k} =‎ ‎‎\sum‎‎‎_{‎gh = k‎}‎ ‎\lambda‎_{‎g}‎\mu‎‎_{‎h}‎‎$‎‎‎
We denote this algebra by ‎$‎‎‎F[G]‎‎$‎.
Definition ‎: The algebra ‎$‎‎‎F[‎G‎‎]‎‎$‎ is called the group algebra of ‎$‎G‎$‎ over ‎‎$‎F‎$‎.
‎So, my questions:
‎

1: Should we first prove the group's properties؟ What conditions should be considered for group multiplication?
2: I did not understand how we got a vector space on ‎$‎‎‎F‎$‎ with base‎ ‎$‎‎‎G‎$‎.
3: Do ‎you‎ guide how ‎this ‎set‎ became an ‎‎$‎F‎$‎- ‎algebra?‎

‎
 A: You can define it so that everything works and you don't really need to check anything.
For example, you can say that you want $F[G]$ to be a vector space over $F$ with a basis consisting of elements indexed by $G$. So basically define the elements to be finite sums of the form $\sum_{i=1}^n f_ie_{g_i}$ where the $g_i$ are elements of $G$. Formally, you should consider functions $\phi: G \to F$ where $\phi(g) = 0_F$ for all but finitely many $g$: do you see why these are the same?
Then there's a canonical vector space structure built in, defined by $(f,f'e_g) = ff'e_g$ then extending by linearity, where $f,f' \in F$ and $g \in G$.
But then you can define a multiplication by saying that $e_ge_h = e_{gh}$ and then just extend by $F$-linearity, To do this, note that each element of $F[G]$ is describable by a finite sum $\sum_i f_ie_{g_i}$. To multiply this with another one, just use distributivity and then the rule $e_ge_h = e_{gh}$. Then you get a multiplication, and you can check it satisfies the axioms of an $F$-algebra.
A: Perhaps the following approach, which lends itself to generalizations, could help you understand the concept.
Consider the set $\mathfrak{A}$ of the functions $f : G \to F$ such that the support $\{ g \in G : f(g) \ne 0 \}$ is finite. Define on $\mathfrak{A}$ pointwise addition (i.e. $(f_{1} + f_{2})(g) = f_{1}(g) + f_{2}(g)$, where the support of $f_{1} + f_{2}$ is contained in the union of the supports of $f_{1}$ and $f_{2}$) and convolution as the product
\begin{equation}
(f_{1} * f_{2})(g) = \sum_{x, y \in G, xy = g} f_{1}(x) f_{2}(y),
\end{equation}
where the sum makes sense, as the support of $f_{1} * f_{2}$ is contained in the product in $G$ of the supports of $f_{1}$ and $f_{2}$.
This is the same algebra of you definition, where the role of the basis element $g$ is taken by the function
\begin{equation}
\delta_{g}(x) =
\begin{cases}
1 & \text{if $x = g$,}\\
0 & \text{if $x \ne g$.}
\end{cases}
\end{equation}
A: *

*This can be done for any group, so it doesn't matter what the group's properties are.

*The construction of such a vector space is done by considering functions $f:G\to F$ where $f(g)=0$ for all but finitely many $g\in G$ with pointwise addition and scalar multiplication. Then we can write $f$ using the notation $\sum_{g\in G} f(g)g$.

*It's a ring that contains $F$, hence an $F$-algebra. (To see that it contains $F$, let $e$ be the identity of $G$. Then $\alpha \mapsto \alpha e$ gives the inclusion of $F$ into $F[G]$.

