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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Generators for $\mathbb{C}[G](1-g) \cap \mathbb{C}[G](1-h)$ given by relations in $\langle g,h \rangle$?

Let $G$ be a group and $g,h \in G$ two elements. I want to describe a set of generators for the left ideal $\mathbb{C}[G](1-g) \cap \mathbb{C}[G](1-h)$. Suppose we have a relation $$g^{n_1}h^{m_1}g^{...
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Isomorphism between $\overline{\mathbb{C}[G \times G]}$ and $\overline{\mathbb{C}[G]} \otimes_{min} \overline{\mathbb{C}[G]}$

I'm currently reading through a paper and it has a sort of throwaway line that says that we have the following isomorphism at our disposal: $\otimes_{k=1}^{n} \overline{\mathbb{C}[G]} \simeq \overline{...
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Inverse limit and crossed product

Let $p$ be a prime and $H$ be a uniform, pro-$p$ group. Then the Iwasawa algebra $\mathbb{F}_p[[H]]$ can be seen as the $I$-adic completion of the group algebra $\mathbb{F}_p[H]$ for $I$ the ...
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Powers of a unit in the group algebra

Let $\mathbb{K}[G]$ be a group ring over some field $\mathbb{K}$, where $G$ is a torsion free group. Let $u\in\mathbb{K}[G]$ be a unit and a central element, and assume that $u$ has vanishing constant ...
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Why is an abelian group an algebra over $GF(2)$?

In this MO answer, Geoff Robinson writes:(Punctuation error corrected) Consider a group homomorphism $\phi:H\to A$, where $A$ is an Abelian group. Let $R$ be the group ring $GF(2)[A]$. Consider $\phi$...
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Calculations in a group ring with computer algebra system

I am currently trying to read the paper Constructions of difference sets by Applebaum et al. Unfortunately, there are some very elaborate calculations. For example, in example 1.13 in the group ring $\...
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Polynomials with exponents in numerical semigroup

When thinking about monoid algebras (semigroup rings), I came across the following: Let $S$ be a numerical semigroup, i.e., an additively closed subset of $\mathbb{N}_0$ containing $0$ such that $\...
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Express a function $g$ in the vector space $\mathbb{Z}/3$

I have a question I can't seem to solve. It is stated as follows: Recall that $\mathbb{Z}$ denotes the ring of integers, and $\mathbb{Q}$ is the rationals. We let $F[x]$ denote the polynomial ring in ...
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No onto map between the algebras.

I want to prove that there does not exist a surjective group algebra homomorphism from $FS_5$ (the group algebra (or group ring) of the symmetric group, $S_5$, over the field $F$) to $M_2(F)$, where $...
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Direct summand of a group algebra.

I want to show that the matrix ring $M_2(\mathbb{F_7})$ can never be the summand of group algebra $\mathbb{F}_7S_5$, where $S_5$ is the symmetric group on $5$ symbols. To be more precise, I want to ...
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Could the discovery of a counterexample to the Unit Conjecture change mathematicians’ understanding of spinors in general?

Recently it has been reported that Giles Gardham has found a counterexample to the Unit Conjecture for group rings, as given in https://arxiv.org/abs/2102.11818 (and also in a popular article https://...
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Tensor products of group algebras

We know that if $G$ and $H$ are finite groups and $F_p$ is a field of characteristics $p$, then $$F_pG\otimes_{\mathbb{F}_p} F_pH\cong F_p(G\times H).$$ Here $\otimes_{\mathbb{F}_p} $ denotes the ...
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Module Over the Group Algebra of a Ring

I'm aware that if $K$ is a field, $V$ a $K$-vector space, and $G$ is a group, then a $K$-representation of $G$ in $V$ is the same thing as a $KG$-module, where $KG$ is the group algebra of $G$ over $K$...
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Why that element belongs to FC subgroup?

Let $\Delta$ denote Finite Conjugate subgroup of group G. K - a field Then The photo above is from Passman book "Infinite group rings". I dont understand why elements $u_{i}$ belong to $\...
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Isomorphism Between Cyclic group algebra and Quotient Polynomial Ring

Let $F$ be a field. I'd like to show that: $a)$ If $V$ is a cyclic group of order $n$, then $$F[V] \cong F[x]/(x^n - 1).$$ $b)$ If $\operatorname{char}(F)\neq 2$, $V=\{1, f, g, fg\}$ is the Klein ...
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Group Algebra of Direct Product is Tensor Product of Group Algebras

Given groups $G$ and $H$, I want to show that $R[G \times H] \cong R[G] \otimes R[H]$. I'm not sure why, but I'm struggling with it. I'd like to map the basis elements from one to the other in the ...
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A problem on group rings from Isaacs Algebra, a Graduate Course

Updated below, but still not sure of my approach. I am trying to solve problem $13.6$ from Isaacs' Algebra, a Graduate Course. I am probably not understanding the question, but here is my attempt. A ...
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Common multiples in a group ring

Consider the group ring $\mathbb{Q} F_2$ with rational coefficients over the free group in two generators ($a$ and $b$). It is not an Ore-Domain: While it does not contain any zero-divisors, for ...
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Show that $N=g_1 + g_2 + \cdots g_n$ is in the center of the group ring $RG$

There is a similar question here which I can understand but there is no mention of conjugacy class in my question so I'm somewhat confused. Let $G=\{g_1, g_2, ..., g_n \}$ be a finite group, show ...
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Is is it possible for the augmentation ideal of group ring $RG$ to be essential? [closed]

A left ideal $I$ of ring $R$ called essential, if $I\cap J\not =0$ for every nonzero ideal $J$ of $R$. Is is it possible for the augmentation ideal of group ring $RG$ to be essential? (You can assume ...
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A question about augmentation ideal in group algebra.

I am new in group rings, and I would like to know more about augmentation ideal. Let $FG$ be a group ring and $Ker(\epsilon)$ be the augmentation ideal. How can we show that $\frac{FG}{Ker(\epsilon)}\...
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Is it true that primitive idempotents of $kS_n \otimes_k kS_m$ are just tensor of idempotents of $kS_n$ and $kS_m$?

I was working over a general field $k$ of characteristic $0$. For now, just think $k$ as the field of complex number. Let $n$ and $m$ be some natural numbers. Let $kS_n$ be the group algebra of $S_n$, ...
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A little bit confused about group rings

According to Wikipedia: Let G be a group, written multiplicatively, and let $R$ be a ring. The group ring of $G$ over $R$, which we will denote by $R[G]$ (or simply $RG$), is the set of mappings $f : ...
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group algebra and its center

Let F be a field and G group so the group algebra is defined as: $F[G] = {\sum_{g\in G} c_{g} g: c_{g} \in F, g\in G}$ take the element $a= \sum_{g \in G} g$, clearly $a\in G$. Can I say that a is in ...
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Groups with isomorphic group rings have the same homology

I am trying to do exercise 9.19 in Rotman's Intro to Homolgical Algebra. Admittedly, I have skipped much of the content leading up to it (which is probably causing my issues). Here's where I'm at: Let ...
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Isomorphism of group rings

I have to prove that $G=\mathbb{Z}/6\times\mathbb{Z}/2$ and $H=\mathbb{Z}/12$ are not isomorphic, but the group rings $\mathbb{C}[H]$ and $\mathbb{C}[G]$ are isomporphic. First part was quite easy. ...
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$\mathbb{Z}_{2}G$ as the class of finite subsets of $G$

Let $G$ be a group. Then the group ring $RG$ is defined as $\sum_{g\in G} R$ (a copy of $R$ for every $g\in G$) and a typical element is of the form $ \sum_{i=1}^{n} r_{g_{i}}g_{i} $. Now if $R= \...
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What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (like a polynomial ring)?

I don't think there is much for me to elaborate beyond the title question: "What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (...
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Show that $K [\Bbb Z]$ is an integral domain.

Let $K$ be a field. Then show that the group ring $K [\Bbb Z]$ is an integral domain. How do I show that $K[\Bbb Z]$ contains no divisor of zero? Any help will be highly appreciated. Thanks in ...
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Same group ring implies same group

is there any example, s.t. $K=\mathbb{C}$ is a field and we have two non-isomoprhic finite groups $G$ and $H$, s.t. the group ring $K[G]$ is isomorphic to $K[H]$ as a $K$-algebra? The idea of this ...
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Example of bialgebra

The following can be found in Montgomery, Hopf Algebras and their Actions on Rings, example 1.3.2. Let $G$ be a group and $B=kG$ be its groups algebra, where $k$ is a field. Then $B$ is a bialgebra ...
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Center of a group ring?

Let $R$ be a ring with unity and let $G$ be a group. Then what is the center of the group ring $R(G)$? I feel that the center $Z(R(G))$ is $R(Z(G))$. I assumed if $x=\sum_{i=1}^{n} r_i g_i$, then it ...
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How is ${}_{kG}\mathcal{M}$ a tensor category?

Let $G$ be a monoid and $k$ be a commutative unital ring. We consider the algebra $$kG = \left\{\sum_{g \in G}' \alpha_g g: \alpha_g \in k\right\}$$ with its usual operations. Some notes I'm reading ...
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Bijection between $KG$-modules and $K$-representations of $G$

Let $K$ be a field and $G$ be a group. Consider the group algebra $KG = \{\sum_{g}'\alpha_gu_g: \alpha _g \in K\}$ where $\{u_g: g \in G\}$ are a $K$-basis. Consider the following operations: $$f: \{K-...
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$Z\pi \cong Z\oplus Z$ and so $P \cong Z$ [closed]

If $\pi = Z$, then the augmentation ideal $P$ is projective and $0 \to P \to Z\pi \to Z \to 0$ is a projective resolution. Here we know that $P$ has basis $\mathbb{Z}-$ ${0}$ Then how to show that it ...
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Computing the Group Ring $k[\mathbb Z / n \mathbb Z]$ for a Field $k$ of Characteristic $0$

Consider a field $k$ of characteristic $0$ and a positive integer $n.$ In the proof of Theorem 4.19 of Polytopes, Rings, and K-Theory by Bruns and Gubeladze, it is stated that we have an isomorphism $...
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Prime ideals in group ring

Let $R$ be a ring of characteristic $p$ (can assume $R=\mathbb{F}_q$ if it makes things significantly easier) and $G$ be a finite group. Do we know anything about the prime ideals in $R[G]$? I've ...
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On structure of the Iwasawa algebra

Let $\Gamma_0 = \mathrm{Gal}(\mathbb{Q}_p(\zeta_{p^\infty})/\mathbb{Q_p})\cong \mathbb{Z}_p^*$, $\Gamma_0\supset\Gamma_n\cong1+p^n\mathbb{Z}_p$ for $n\geq 1$. Then we have $\Gamma_0=\Delta\times\...
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Is problem of finiteness in the sense of von Neumann field FG algebras solved for any class of groups?

there is an open problem: Does for every field $F$ of characteristic prime and every group $G$, the group ring $FG$ finite in the sens of von Neumann? Is this problem solved for any class of groups? I ...
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Element of group ring not invertible, but invertible in bigger group ring.

I have thought about such problems: Is there a field $F$ and a group $G$ such that there exists element $x$ of group ring $FG$ which is not invertible in $FG$, but is invertible in $KG$, where $K$ is ...
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Scalar extension of group algebras

Let $R$ be a commutative unital ring, let $G$ be a group and let $R[G]$ its group $R$-algebra. It is true that $\mathbb{Z}[G]\otimes_{\mathbb{Z}}R\simeq R[G]$? If not, there are conditions on $R$ and ...
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How to show $\operatorname{Ext}^1(I, \mathbb{Z})\cong\mathbb{Z}/2$ for $\mathbb{Z}[C_2]$?

I am trying to become better acquainted with $Ext^1$ and thought the best place to start is with actual computations. This was an exercise in some brief notes I found and wanted to see if my thinking ...
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The category of finite dimensional right $KG$-modules is equivalent to the category of finite dimensional representations of a quiver $Q$

Let $K$ be an algebraically closed field and $G$ be a finite group such that $|G|$ is not divisible by the characteristic of $K$ (so that Maschke's theorem can be applied). Let $Q$ be the quiver ...
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Does every finite dimensional algebra contains a semisimple subalgebra?

As in title - does every finite dimensional algebra over field K contains a semisimple (or semiprime, as in finite dimension it is the same) subalgebra? Due to coronavirus I dont have my notes from ...
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Structure of ideals in $\mathbb{F}_q[G]$

Let $\mathbb{F}$ be a finite field of characteristic $p$ and let $G$ be a cyclic group of order $p^n$. I read in a paper that all ideals of the group ring $\mathbb{F}[G]$ are of the form $I_n^j$ where ...
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Dimension of center of k[G]/rad k[G] where characteristic of k divides the order of G.

Let G be a finite group and consider k[G] where k is a field. In the scenario where char(k) divides |G|, how can one show that the dimension of Z(k[G]/rad k[G]) is strictly less than dimension of Z(k[...
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Let $x^2=y^2=1$ and $xy\neq yx$. There are $\binom{2n}{n}$ expressions of length $2n$ in $x$ and $y$ that are equal to $1$.

This question is motivated by this link. The statement is as follows. (Edit: Even if there are already two great answers, I would love to have a couple more answers. Especially, I would like to see ...
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Why is the constant term of $(1+x+y+xy)^n$ equal to $\frac{1}{2}\binom{2n}{n}$?

If we define this: for any $x,y$ such that $x^2=y^2=1,xy\neq yx$, express in terms of $n$ the constant term of the expression $$f_{n}=(1+x+y+xy)^n\,.$$ I guess this result is $\dfrac{1}{2}\binom{...
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Szamuely's "Galois groups and fundamental groups" exercise I.7

I am working through the exercises in Tamás Szamuely's book "Galois group and fundamental groups". Exercise 7 from the first chapter is the following. Let $k$ be a field and $\bar{k}$ be a fixed ...
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Invertibility in the Complex Group Ring

Let $\Gamma$ be a discrete group, and let $\Bbb{C}[\Gamma]$ be the associated complex group ring. If $\sum_{g \in \Gamma} a_g g$ represents some element in the group ring, where all but finitely many ...

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