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Let $G$ be a group. Let $K$ be any field and consider its group algebra $K[G]$. It is known that if $G$ is a finite group then $K[G]$ is never a field. Or even if $G$ is not finite: if $G$ has at least one non-trivial torsion element, then $K[G]$ is not a field.

Question: Do there exist infinite groups $G$ such that $K[G]$ is a field or a division ring?

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1 Answer

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When $|G|>1$ the augmentation ideal is nontrivial, so it is never a simple ring, much less a field or division ring.

The augmentation ideal is the kernel of the ring map $\sum_{g\in G} r_gg\mapsto \sum_{g\in G}r_g$. It is nontrivial, for example, because $1-g$ is in it for any nonidentity element $g\in G$.

So, if $K$ is a field, the only way for $K[G]$ to be a division ring is if $|G|=1$.

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