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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9
votes
1answer
389 views

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 ...
54
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3answers
2k views

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 ...
40
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3answers
2k views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then $k[G]...
10
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1answer
438 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
9
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1answer
958 views

Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ is a nilpotent ideal ($p$ is a prime, $G$ is a $p$-group)

Let $p$ be a prime and let $G$ be a finite group of order a power of $p$ (i.e., a $p$-group). Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ (to be read as $\left( \...
8
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1answer
895 views

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 ...
3
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1answer
204 views

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 ...
2
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1answer
57 views

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 ...
1
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1answer
63 views

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 ...
6
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2answers
183 views

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 (...
9
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1answer
777 views

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 ...
7
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1answer
422 views

How to recover the integral group ring?

I would like to solve the following exercise: Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
7
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1answer
1k views

Artin-Wedderburn decomposition of a particular group ring

I am trying to do a question from an algebra qualifying exam: Decompose the group ring $\mathbb{F}_5[S_3]$ as a product of simple rings. By Maschke's theorem since $\mathrm{char}(\mathbb{F}_5) \...
3
votes
1answer
706 views

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 ...
10
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2answers
193 views

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 $F$ ...
7
votes
1answer
273 views

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 ...
5
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1answer
205 views

Can the averaging of a linear arrow of modules over a group algebra be described functorially?

Wikipedia outlines a neat proof of Maschke's theorem from the module theoretic perspective. The fundamental idea seems to be "averaging". Proposition. Write $U:\Bbbk G$-$\mathsf{Mod}\to \Bbbk$-$\...
9
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3answers
267 views

Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere ...
8
votes
1answer
276 views

find a special element in group algebra

Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual ...
3
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1answer
257 views

units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic

How would I be able to describe all units of the group ring $\mathbb{Q}(G)$ where $G$ is specifically an infinite cyclic group?
3
votes
1answer
115 views

Computing the Jacobson radical of $F[G]$ with $char(F)=p$ and $G$ finite with a normal $p$-Sylow subgroup

Let $G$ be a finite group and $F$ be a field of prime characteristic $p$. Suppose further that $G$ has a normal $p$-Sylow subgroup $N$. Question: Is it true that the Jacobson radical of the ...
3
votes
1answer
783 views

How to calculate inverses in the group algebra

Is there an algorithm to calculate the inverse of an element in the group algebra? For example, does the element $(1 2 3) + 2 . (1 2)(3 4)$ in the group algebra $\mathbb{C} S_{4}$ have an inverse, ...
2
votes
1answer
91 views

Conditions for free/projective/flat module over a groupring

Let $H \subset G$ be a subgroup of the group $G$. When is $\mathbb{Z}[G]$ a free/projective/flat $\mathbb{Z}[H]$-module? If $\mathbb{Z}[G]$ is a free $\mathbb{Z}[H]$-module then there is a $n \in \...
7
votes
2answers
135 views

Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I ...
3
votes
1answer
278 views

Units in finite polynomial rings

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $...
2
votes
2answers
102 views

Center of Group algebra finitely generated

I am going through a proof in Jean-Pierre Serre's german version of "Linear representations" and have the following theorem here. Let $\rho$ be an irreducible representation of the finite group G of ...
2
votes
2answers
191 views

Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
2
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1answer
102 views

Introductory text for Group Rings

Is there any other text books on Group rings except The algebraic structures of Group Rings by D.Passman. This book is really good but it will help if I know about other books on the topic too. Thanks!...
1
vote
1answer
56 views

Is $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID?

As the title suggests, I'm interested whether $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID or not. Assume $p$ is prime. My feeling is that it is a PID, since $\mathbb{Z}/(p)$ is cyclic an morally if an ideal ...
0
votes
1answer
76 views

Ring isomorphism for $k(G \oplus \mathbb Z )$ with $G$ torsion-free and abelian

Let $k$ be a field and $G$ be a torsion-free abelian group. Then $k[G]$ is an integral domain. If we denote its field of fractions by $F = k(G)$, is it true that $k(G \oplus \mathbb Z )\cong F(X)$? ...
0
votes
2answers
74 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} $...
0
votes
2answers
115 views

If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$

Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism $$ \varphi(1) = (1,0), \...
0
votes
1answer
63 views

Looking for a good name for this “Quantisation Regime”

This is an attempt to salvage the wreckage of my hope to motivate (finite) quantum groups a lá this question. Let $\{S_i\}_{i=0}^n$ be a family of finite sets and let $\varphi$ be a map $$\varphi:...