# 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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### Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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### The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
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### Is there an 'easy' way to calculate $K_0(\mathbb{Z}[C_p])$?

For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal ...
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### Decomposing $\mathbb{F}_p[G]$ ($G$ finite) into products of matrix rings over fields

I have recently begun learning about group algebras over finite fields but am still a little uncertain about these guys. So I was looking for some clarification and verification. Consider the ...
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### Jacobson radical of integral group ring ZS3

I want to know whether Jacobson radical of integral group ring ZS3 (S3 is symmetric group) is known or not. Please help me. Tell me about any reference. Also if anyone know about the maximal ideals ...
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### Equivalent condition for normalizer problem

Let $\cal{U}$ be the unit group of group ring $\Bbb{Z}G$ then the Normalizer Problem (NP) states that $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}=\cal{Z(U)}$. Now why (NP) is equivalent to saying that ...
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### What is the augmentation ideal of $\Bbb{Z}S_3$

I know that $\Delta_{\Bbb{Z}}(G)$ is the $\Bbb{Z}-$ module generated by elements of form $\{g-1\ \forall\ g\in G\}$. But how do we find them or what it looks like. I was thinking about finding aug ...
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### Calculation in a Group Ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
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### how to prove intersection of normal subgroups controls the ideal

I have to prove the following: Let $I$ be an ideal of $K[G]$ ($K$ is a field, $G$ is a multiplicative group, $K[G]$ is a group ring) and let $H_1,\ldots,H_n$ be normal subgroups of $G$, each ...