# 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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### Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
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### Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?

Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...
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### Which element of $\mathbb{Q}C_5$ will be mapped onto $(0,4)\in \mathbb{Q} \oplus \mathbb{Q}(\zeta)$

I know that the rational group algebra of $C_5 = \langle g : g^5=1\rangle$ is $\mathbb{Q}C_5 \cong \mathbb{Q} \oplus \mathbb{Q}(\zeta)$ where $\zeta$ is a root of $x^4+x^3+x^2+x+1$. But I was ...
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### Morita equivalence between $\mathbb{C}[G]$ and $\mathbb{C}[H]$?

What we can say about two groups G and H when their group rings, $\mathbb{C}[G]$ and $\mathbb{C}[H]$, are Morita equivelent?
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### Structure of $\mathbb{Q}S_3$

I have an exercise to show that $\mathbb{Q}S_3 \cong \mathbb{Q} \oplus\ \mathbb{Q}\ \oplus\ M_2(\mathbb{Q})$ , where $M_2(\mathbb{Q})$ is ring of $2$ by $2$ rational matrices and $\mathbb{Q}S_3$ is ...
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### prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$

I want to prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$, where $C_2=<x| x^2=1>$, where $\mathbb{Z}(G\times C_2)$ and $(\mathbb{Z}G)C_2$ are integral group rings and I am looking for ring ...
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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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### group ring Z(G) is isomorphic to tensor algebra T(G)

Well, I've been learning Tensor algebra and related topics. So, I wonder tensor algebra is related to somehow familiar with me, so called group ring. $\mathbb{Z}$: a ring of integers. Let G be a ...
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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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### Difference between $\mathbb{Z} G$-module and $G$-module

I am studying Group Cohomology, but I am a little confused about $\mathbb{Z} G$-module and $G$-module. Some text uses the $G$-module for group cohomology, but I thought group cohomology of $G$ is ...
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### How do I view $k$ as $kG$-module?

How could I view $k$ as $kG$-module? Does $kG$ acts on $k$ by $rg\cdot r'=rr'$, just ignore $g$? Could we do that?
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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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### Projective but not free module over groupring

Let $G$ be a nontrivial finite group and consider the groupring $\Bbb QG$. My question is whether we can find a module over $\Bbb QG$ that is projective but not free?
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### Show $D = \delta_1 +\cdots+ \delta_m$ is in the center of the group ring $RG$ .

Let $\Delta$ = {$\delta_1,\dots, \delta_m$} be a conjugacy class in a finite group $G$. Prove that the element $D = \delta_1 + \cdots + \delta_m$ is in the center of the group ring $RG$ . Hint: ...
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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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### Decomposition of group algebra $\mathbb{C}[G]$ [duplicate]

Let $G$ be a finite group. By Wedderburn theorem, we see that $\mathbb{C}[G]$ is product of matrix algebras. "Linear Representations of Finite Groups" by Serre has an explanation for this result ...
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### Is this module over this group algebra projective?

Assume that $G$ is a finite group. Let $k$ be a field. Let $\varepsilon$ be the augmentation $kG\rightarrow k$. Consider the following map $\varepsilon\otimes id:k[G]\otimes_k k[G]\rightarrow k[G]$...
Let $G$ be a finite group and $F$ a finite field with co-prime orders. Show that in the group algebra $F[G]$, if $x \in F[G]$ is not a left zero divisor then it is invertible. Thoughts so far: By ...
Suppose $R$ is some commutative ring and $G$ a finite group so that $R[G]$ is the usual group ring. If $M$ is some $R[G]$-module, then we can inject $M\hookrightarrow M\oplus R[G]^n$ for some $n\geq 1$...