# 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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### How "exotic" are non-local directly irreducible modular group algebras?

Let $G$ be a finite group and $R = \Bbb F_{p^k} G$ a modular group algebra ($p$ divides $|G|$). I would like to know when $R$ is a directly indecomposable ring. It is quite well-known that $R$ is ...
1 vote
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### Has anyone studied differential equations on $\mathbb{Z}F_n$ defined using Fox derivatives?

I am looking for a reference, if it exists, to the study of differential equations defined using Fox derivatives over the group ring, say, of a free group. Is this a topic which has been studied ...
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### Module structure on $\mathbb{Z}$ over the Groupring $\mathbb{Z}[G]$

I would like to know how exactly the module structure on $\mathbb{Z}$ as a module over the group ring $\mathbb{Z}[G]$ works. Obviously, we take just the usual addition on $\mathbb{Z}$, but how does ...
1 vote
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### Is there a way to determine when an abelian group $(M,+)$ admits a $R[G]$-module structure, where $R[G]$ is a group-ring.

As the title says, are there any relevant theorems, which determines when, given an abelian additive group $(M,+)$, we can determine whether $(M,+)$ admits a $R[G]$-module structure, for a unital ring ...
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### Group ring zero divisor example.

Here is the question I am trying to understand its solution: Give an example of a zero divisor in the group ring $\mathbb Z(C_3)$ where $C_3$ is the cyclic group with 3 elements. My professor gave us ...
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1 vote
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### Image of augmentation ideal of a group ring under ring homomorphism.

Let $G,H$ be two finite $p$-groups and $f:\mathbb{F}_p[G]\to \mathbb{F}_p[H]$ a ring homomorphism of mod $p$ group rings. Is it always true that $f(I_G)\subset I_H$, where $I_G,I_H$ are the respective ...
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### Description of projective $\mathrm{SL}_{2}(\mathbb{Z})$-modules

I am working my way through Ken Browns book on the cohomology of groups, and in particular chapters 8 and 9 on finiteness conditions, and Euler characteristics. Most of the concepts in chapter 9 (such ...
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### Recovering original definition of group cohomology from Ext definition

I've recently been studying group cohomology, the original definition I learned was that of Ext, where $H^n\left(G, M\right)= \text{Ext}_{\mathbb{Z}G}^i\left(\mathbb{Z}, M\right)$. I then read a ...
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### Is the direct sum preserved under isomorphism

I have edited this based on discussions in the comments: This really is more of a sanity check and I couldn't seem to find an answer to anything similar anywhere, so apologies if this has already been ...
1 vote
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### Decompositions of a group algebra

I am a math student just starting to know group algebra. Let $FG$ be a group algebra in which $F$ is a field and $G$ is a group. I am particularly interested in the results of the decomposition of $FG$...
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### About the definition of group algebra

In several texts it is standar to define the group algebra $\mathbb{C}[G]$ of a group $G$ as $$\mathbb{C}[G]:=\left\{ \sum_{x\in G}c_{x}x \colon c_{x}\in \mathbb{C} \right\}.$$ Now, these ...
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### Representation of $\mathfrak{S}_p$ over $\mathbb{F}_p$

I'm learning modular representation theory from the 3rd part of Serre's book. Through the process called "reduction mod. $\mathfrak{m}$", we obtain representations in positive characteristic ...
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### $\mathbb{Z}$ as $\mathbb{Z}G$ - module.

Let G group. Recently i read that $\mathbb{Z}$ can be considered as $\mathbb{Z}G$- module with the trivial action, such that : $$\forall x\in \mathbb{Z}G,\ z\in \mathbb{Z}:\quad x\cdot z:=z$$ But if ...
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### Is (3) a prime ideal of $\mathbb{Z}$ $(\sqrt{-17})$

First, I had to prove that $\mathbb{Z}$ $(\sqrt{-17})$ is not a UFD which was done by showing that in general $\mathbb{Z}$ $(\sqrt{-n})$ for $n \geq 3$ is not a UFD. Subsequently, I was asked if (3) ...
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### Lie algebra structure on $\mathbb{C}[G]$

Let $G$ be a finite group. The group algebra $\mathbb{C}[G]$ has a Lie algebra structure inherited by its associative algebra structure, which is given by $$[g,h]=gh-hg.$$ It is not too hard to check ...
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### Group Ring/Algebra and Functions of Positive Type

Given a group $G$, we can form its complex group ring/algebra $\Bbb{C}[G]$, which either be described as the set of all finite formal sums of group elements with complex coefficients or as all the ...
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### Homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$

Show that there are at most $4$ ring homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$. Here is what I did : We know that ring homomorphisms send inversible elements to ...
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### Doubt about invertibility in Group Algebra over a cyclic group

Let $C=\langle c \rangle$ denote the cyclic group of order $2$ written multiplicatively, i.e. $C$ is a copy of $\{\pm 1\}$ with the usual product, and let $k$ be a field of characteristic different ...
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