Let $G$ be a finite group. Then the rational group algebra $\mathbb{Q}[G]$ has a wedderburn decomposition of the form $\prod\_i M\_{n\_i}(D\_i)$ where each $D\_i$ is a division algebra. 


My question is: for which $G$ do we have $n\_i=1$ for all $i$? In other words, for which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras? 


Clearly, this is true when $G$ is abelian. Another example is $G=Q\_8$, the quaternion group of order $8$. Are there other examples? Is there a complete classification of such groups? If so, can you provide a reference?



For which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras?
The groups you are looking for are precisely those for which the group algebra $\mathbb Q[G]$ does not contain nonzero nilpotent elements. These groups have been classified by Sehgal in [here](http://link.springer.com/article/10.1007%2FBF01168879). The finite groups which have this property are the abelian ones and the Hamiltonian groups of order $2^mt$ (where $t$ is odd), such that $2$ has odd order modulo $t$.


