Let $A$ be an associative algebra over a filed $k$.


**Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?**


I am particularly interested in the following cases:


1) When $k=\mathbb{Q},\mathbb{R}$ or $\mathbb{C}$. 


2) $A=M\_n(k),$ the matrix algebra or a subalgebra of matrix algebra.


PS: I am not sure whether this question is of research level or not. If anybody thinks that this is not proper here please give the references and then vote to close. I have searched it in general but could not find any answer. Thanks in advance.



Relation between Associative algebra and group algebra
If $k$ has characteristic zero and $A$ is finite-dimensional, by Maschke's theorem it must be semisimple. If $k$ is in addition algebraically algebraically closed and $A \cong k[G]$, then $A \cong \prod\_i M\_{n\_i}(k)$ where $n\_i$ are the dimensions of the irreducible representations of $G$. So in this case the question reduces to the following purely group-theoretic question:



> 
> When is a multiset $n\_i$ of positive integers the multiset of dimensions of the irreducible representations of a finite group over $k$?
> 
> 
> 


Some miscellaneous necessary conditions are that 


* one of the $n\_i$ must be equal to $1$ (the trivial representation),
* each $n\_i$ must divide $\dim A = |G| = \sum n\_i^2$,
* the number of $n\_i$ equal to $1$ (the size of the abelianization of $G$) must also divide $\dim A$,
* if $\dim A$ is prime (so $G$ must be cyclic), then each $n\_i = 1$.


The first condition rules out $M\_n(k)$ for $n \ge 2$. The second condition rules out lots of examples, such as $k \times M\_2(k)$. The third condition rules out lots more examples, such as $k \times k \times M\_3(k)$. And so forth. I can't imagine there are any useful sufficient conditions; there are a lot of finite groups and they can be very complicated. 


