# Questions tagged [group-rings]

A group ring $R[G]$ is a ring constructed from a group $G$ and ring $R$. A special case of this construction is group algebra, which occurs naturally in representation theory.

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### Isomorphism between $I_G/I_G^2$ and $G/G'$

Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it. $G$ is ...
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### How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
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### Compute the Jacobson radical of the group ring $\mathbb{F}_2S_3$.

Compute the Jacobson radical and the maximal semisimple quotient of the group ring $\mathbb{F}_2S_3$ of the symmetric group on three letters over the field with two elements, and compute the same ...
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### Which rings arise as a group ring?

Let $R$ be an arbitrary associative ring with identity. When does there exist a group $G$ and a field $F$ such that $F[G] = R$? Do we obtain more rings as $F[G]$ if we loosen the condition that $F$ ...
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### Positive Elements of $\mathbb{C}G$: as functionals versus as elements of the C*-algebra

I might have thought about this problem a little longer but am quite confused so said I would put this question to the good people here... Consider a finite group $G$ or rather the algebra of ...
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### GAP: how to work with elements of the group ring of the symmetric group $\mathbb{C}[S_k]$?

In GAP, working with elements of the symmetric group $S_k$ is straightforward. E.g. one can write (1,2)*(2,3); to obtain (1,3,2). Is there a similar ...
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### Description of the group $1+J(FG),$ where $J(FG)$ is jacobson radical of the group ring $GF.$

My group is $G=(\mathbb{Z}_3\times\mathbb{Z}_3)\rtimes\mathbb{Z}_3$ which is non abelian group of order $27.$ Now my problem is whether the group $1+J(FG)$ is abelian or non-abelian and what is its ...
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### Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?

I know for two groups $G, H$ (not necessarily finite) we have $R[G\times H]\cong (R[G])[H]$, but I was wondering if we had a similar statement for rings $R,\,S$. In other words, if $R,\,S$ are two (...
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### If $G$ is an infinite group, then the group ring $R(G)$ is not semisimple.

Let $R$ be a ring and $G$ an infinite group. Prove that $R(G)$ (group ring) is not semisimple. My idea was to suppose it is semisimple, then $R(G)$ is left artinian and $J(R(G))=0$. I was trying to ...
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### Quotient of ideal in group ring is isomorphic to abelianization [duplicate]

Let $G$ be a group and $\mathbb Z G$ the group ring over the integers. Let $I$ be the ideal of elements $\sum_{g\in G} n_g g$ with $\sum_{g\in G} n_g = 0$. I am trying to prove that $I/I^2$ is ...
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### Jacobson radical of the integral group ring

I am trying to prove that the Jacobson radical of the integral group ring $\mathbb{Z}G$ for a finite group is zero. Most of what I find on semisimplicity, Jacobson semisimplicity, has to do with ...
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