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.

Filter by
Sorted by
Tagged with
55
votes
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 ...
42
votes
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]...
28
votes
1answer
1k views

When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then we can form the group ring $R(G)$. Note that if $g^{n+1} = e$ then $(e-g)(e+g\...
14
votes
0answers
293 views

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 $G$-...
12
votes
1answer
1k views

Augmentation ideal of the group ring

Let $G$ be a group and $I_G$ be the augmentation ideal of the group ring $\mathbb{Z}G$, i.e. $I_G$ consists of formal linear combinations $\sum n_i g_i$ ($n_i\in\mathbb{Z}$, $g_i\in G$) such that $\...
12
votes
0answers
291 views

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 ...
10
votes
2answers
199 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 $F$ ...
10
votes
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
votes
2answers
338 views

The group ring of a ring.

Let $R$ be a ring. Since $R$ is also a group then we can talk about the group ring $R[R]$. I want to understand this group ring $R[R]$. An element $x\in R[R]$ is written as a finite formal sum $$x=...
9
votes
3answers
269 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 ...
9
votes
1answer
791 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 ...
9
votes
1answer
391 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 ...
9
votes
2answers
127 views

Reference for Kaplansky's proof that in $\mathbb{C}[G]$, $ab=1$ implies $ba=1$

Here on the Wikipedia page for Group Rings, talking about group rings over infinite groups, The case [of a group ring $R[G]$ where $G$ is an infinite group] where $R$ is the field of complex ...
9
votes
1answer
1k 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
votes
1answer
905 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 ...
8
votes
1answer
203 views

Torsion free abelian groups $G,H$ such that $k[G] \cong k[H]$ (as rings) for any field $k$

Let $G,H$ be torsion free abelian groups such that $k[G] \cong k[H]$ for any field $k$. Then is it true that $G \cong H$ ? If this is not true, then what if I change the hypothesis to $R[G]\cong R[H]...
8
votes
1answer
427 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 ...
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 ...
7
votes
2answers
205 views

Isomorphism of group rings over finite cyclic groups

Let $R$ be a commutative ring with unity. For any group $G$, let the group ring be denoted by $R[G]$. If $R[\mathbb Z/(n) ] \cong R [\mathbb Z / (m)]$ as rings , then is it true that $m=n$ ? If that ...
7
votes
1answer
639 views

Example of non isomorphic groups with isomorphic group algebras

Below is the construction of two non isomorphic groups, $G_1$ and $G_2$ such that $KG_1 \cong KG_2$ for any field $K$. (My Doubts lie within.) Consider two groups $Q_1=\langle x_1,y_1,z_1\ |\ ...
7
votes
1answer
279 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 ...
7
votes
2answers
136 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 ...
7
votes
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) \...
6
votes
1answer
595 views

For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?

I'm reading the Atiyah-MacDonald book on Commutative Algebra. At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is: $G$ = finite group, $...
6
votes
1answer
144 views

Finite conjugate subgroup

In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached- What is finite conjugate subgroup of a group? It is not ...
6
votes
2answers
144 views

Product of class sums

Let $C_i$ be the conjugacy classes of a finite group $G$. Consider the class sums $z_i=\sum_{g\in C_i} g$. It is well known that ${z_i}$ form a basis of the center of the group algebra $\mathbb{C}G$. ...
6
votes
2answers
184 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 (...
6
votes
1answer
196 views

Thinking about the group algebra $k[G]$ as functions on $G$

Given a group $G$ and field $k$ one can define the group algebra $k[G]$ in two ways: The underlying vector space of $k[G]$ is the free $k$-vector space on $G$, and the multiplication on $k[G]$ is the ...
6
votes
1answer
121 views

Prove that $u$ is a trivial unit

Let $G$ be a finite group. I want to prove that if $u\in \mathbb{Z}G$ be a torsion unit (i.e. for some $n$, $u^n=1$) of integral group ring $\mathbb{Z}G$ such that $u$ normalizes $G$, then $u$ is a ...
6
votes
0answers
196 views

Finding elements in a group ring

Suppose we have a discrete group $G$, finite or infinite, on which we form the group algebra $\mathbb{F}_2[G]$. Suppose also that we have a map $S$: $$S : \mathbb{F}_2[G] \to \mathbb{F}_2[G]: \sum{...
5
votes
1answer
126 views

Tensor product of group algebras

Let $G,G_1$ and $G_2$ are three abelian groups with group homomorphisms $\phi_i:G\to G_i$. This gives $k$-algebra homomorphisms $k[\phi_i]:k[G]\to k[G_i]$. So we can consider $k[G_i]'s$ as $k[G]$-...
5
votes
1answer
209 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 $U:\Bbbk G$-$\mathsf{Mod}\to \Bbbk$-$\...
5
votes
1answer
158 views

Categories for the working mathematician exercises III 1

I'm currently reading Mac Lane's Categories for the working mathematician and I'm having some trouble with the two following exercises from part III. Find (from any given object) an universal arrow ...
5
votes
1answer
192 views

When is a Hopf Algebra isomorphic to a group ring k[G]?

Let $H$ be a Hopf algebra over a field $k$. What are some nice conditions for when $H$ is isomorphic to $k[G]$ for a finite group $G$? The co-multiplication structure on the group algebra $k[G]$ is ...
5
votes
1answer
467 views

Projective modules over $kG$ equivalent to injective.

Let $k$ be a field and $G$ is finite group. I want to prove that a $kG$ module $P$ is projective iff it's injective. I proved that if module is projective then it's injective. 1) $kG$ is injective ...
5
votes
0answers
63 views

Free resolution of $\mathbb Z\times \mathbb Z$.

For the group of integers $\mathbb Z$, we know that we have a finite free $\mathbb Z[t^{\pm 1}]-$resolution of $\mathbb Z$: $$0\longrightarrow \mathbb Z[t^{\pm 1}] \stackrel{t-1}{\longrightarrow} \...
5
votes
0answers
178 views

No onto map between group algebras $FS_5$ onto $M_6(F)$.

I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $...
5
votes
0answers
76 views

When is $R^{op}\cong R$?

Suppose $R$ is a noncommutative ring. When could I reasonably expect $R^{op}\cong R$? For instance I know group rings have a natural involution, i.e. if $SG$ is the group ring in question then $r\...
5
votes
1answer
83 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
5
votes
0answers
202 views

An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
4
votes
2answers
261 views

Are there homomorphisms of group algebras that don't come from a group homomorphism?

Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces ...
4
votes
2answers
110 views

Existence of nontrivial unit in $\mathbb{Q}[G]$, where $G$ is finite.

Suppose $G$ is a finite group of order $|G|>1$, and $\mathbb{Q}[G]$ is the group ring. I'm curious about an example of a nontrivial invertible element, i.e., one that is not of the form $ag$, with $...
4
votes
1answer
176 views

Generalized Formal Power Series Ring

I came up with the following generalization of formal power series and wonder if anyone knows a reference where this is studied. Let $R$ be a commutative ring with unity and $M$ a commutative monoid ...
4
votes
1answer
621 views

Group ring C[Z/n] and Artin-Wedderburn decomposition

I am trying to answer the following questions, which I assume follow on from eachother each other; Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings. For abelian groups $G_1$, $G_2$, $\...
4
votes
1answer
338 views

For a group ring, finding if a subset is an ideal. [closed]

For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
4
votes
2answers
101 views

Question on Group ring $\mathbb{Q}G$ where $G$ is a finite group

Let $G=\{e=g_1,g_2,.....g_n\}$ be a finite group of order n and let $\mathbb{Q}G$ be the group ring. Let $\sigma=\sum_1^ng_i$. Prove that $\sigma^2=n\sigma$ and deduce that $\mathbb{Q}G$ has a ...
4
votes
1answer
93 views

Help in understanding what is going on in $A[G]$

Ok, so Im given a group $G$ and a ring $A$, and define: $$A[G]=\left\{\sum_{g \in G} f(g) g : f:G \to A, \text{ such that $f$ has finite support} \right\}$$ Define the sum $(+)$: $$\sum_{g \in G} ...
4
votes
1answer
92 views

Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID?

Is $\mathbb{Z}_{p}[G]$ a PID, where $G=(\mathbb{Z}_{p},+)$ is the additive group of the $p$-adics $\mathbb{Z}_{p}$? I am studying a paper where the authors implicitly use that claim, but it is ...
4
votes
1answer
120 views

What is the Wedderburn decomposition of $\mathbb{R}[D_{2n}]$?

I have been looking everywhere and can't seem to find a general formula for the Wedderburn decomposition of the real group ring of the dihedral group ring of order $2n$, $\mathbb{R}[D_{2n}]$. Does ...
4
votes
1answer
137 views

A question about Group Rings

Let $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integers module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, u_1^p=1,\,\,u_{...