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.