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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1answer
24 views

The maps in the $\mathbb Z [\mathbb Z]-$free resolution of $\mathbb Z$

I read in many places that the "usual" free resolution of $\mathbb{Z}$ as a trivial $\mathbb{Z}[\mathbb{Z}]$-module is given as follows $$0 \to \mathbb{Z}[\mathbb{Z}] \xrightarrow{\partial} \mathbb{...
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What is the differential $d^0(f) : C^0(G, A) \to C^1(G, A)$ knowing the general formula?

Romyar Sharifi's Lecture Notes It's from the first differential formula appearing on page 7 and that formula is: $$ d^i : C^i(G, A) \to C^{i+1}(G, A), \\ d^i(f)(g_0, \dots, g_i) = g_0 f(g_1, \dots, ...
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1answer
83 views

Group Rings of Topological Groups and Fields

Suppose $\Bbb{K}$ is a topological field and $G$ is a topological group. Recall that $\Bbb{K}[G]$ denotes the group ring of $G$ over $\Bbb{K}$, which consists of sums of the form $\sum_{g \in G} a_g g$...
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1answer
20 views

Confused about the $K$-algebra homomorphism from a representation

The following is on Page 4 of Representation Theory A Combinatorial Viewpoint by AMRITANSHU PRASAD: I am confused by this equation: if $f$ is an element in $K[G]$ (the group algebra), and $g$ is an ...
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15 views

Partially commutative elements in powers of augmentation ideal

Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
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1answer
33 views

How do I construct the group algebra of a group in GAP?

I tried the following: ...
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65 views

A question about group ring of a group over matrix ring!

Let $S$ be a ring and $G$ a group. We denote by $SG$ the group ring of $G$ over $S$. Let $S=M_n(R)$ be the set of all $n \times n$ matrices over a ring $R$. Is it true $SG\cong M_n(RG)$ (as rings)? ...
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1answer
58 views

Find an example of a group algebra with non-trivial solutions to $x^2=x$.

I am looking for a group-algebra with a non-trivial solution to $x^2=x$. That is to say, a solution with $x\neq 1$ and $x \neq 0$ where $1$ is the identity. We have $x\subset \mathbb{C}[G]$ for some ...
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3answers
32 views

Hom$_{i[k]}(k[G], R) \approx$ Hom$(G,R^*)$

Let $k$ be a field, $G$ a group and $R$ a $k$-algebra (i.e. a ring $R$ with a homomorphism $i : k \rightarrow Z(R)$). The claim is that there is a natural bijection between the set of $k$-algebra ...
2
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1answer
58 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 ...
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7 views

Orders and classes of nilpotency in modular group algebras

Let $p$ be a prime number, $K$ a finite field, $char(K)=p$, $G$ a finite $p$-group and $x\in J(KG)$. $1+x$ is an unit of $1+J(KG)$. My question is what is the connection between the order of the unit $...
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1answer
47 views

Sums of Commutators in a Group Ring

Let $G$ be some group, and let $\Bbb{C}G$ denote the complex group ring over $G$. Let $x,y \in \Bbb{C}G$, and define $[x,y] := xy-yx$ to be a (ring) commutator in $\Bbb{C}G$. Let $K$ be collection of ...
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26 views

Proving that $\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a complete intersection ring

I have found in some sources that the ring $R=\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a local complete intersection ring. I need this result in order to apply a related theorem and I ...
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1answer
38 views

trivial modules of group rings

Let $R=\mathbb{F}_p[D]$ where $D$ is a finite group of order prime to $p$. Let $M$ be any simple $R$-module. If one knows that $H^0(D,M)=0$, is $M=0$? If not, under what further conditions can one ...
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285 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 $G$-...
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1answer
20 views

Showing W=ku+kv has a unique module structure over kG

Let $G=<x> \times <y>$ where $|x|=|y|=p$ and so $|G|=p^2$. Let $k$ be a field of characteristic $p$. Let $W$ be the $k$-span of $v$ and $u$. We wish to show the module structure given by $...
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22 views

Decomposition of $C[G]$ as $C[G]$-module into direct sum of submodules of the form $Ce_{\chi}$

Let C be the complex field, and $G$ be a cyclic group generated by $a$. The group ring $C[G]$ is a $C[G]$-module (over itself) with the module action $C[G]\times C[G] \rightarrow C[G]$ the same as the ...
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1answer
68 views

Extending scalars from $\mathbb{Z}[G]$ to $\mathbb{Q}[G]$

Let $G$ be a finite group. Let $M$ and $N$ be finitely generated $\mathbb{Z}[G]$-modules such that $M$ is free as a $\mathbb{Z}$-module. Suppose that $\mathbb{Q}\otimes_\mathbb{Z}M$ and $\mathbb{Q}\...
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39 views

Group ring $R[G]$ semisimple if and only if $J = J^2$?

Let $G$ be an abelian group and let $R$ be a commutative ring and consider the group ring $R[G]$ of finite formal linear combinations of elements of $G$ with coefficients in $R$. Let $J = (1 - g ~|~ g ...
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28 views

Group algebra is domain iff doesn't contain nonzero element whose square is 0

Let $G$ be a torsion-free group and let $K$ be a field. I have to prove that the group algebra $KG$ is an integral domain if and only if it doesn't contain a nonzero element whose square is equal to 0....
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62 views

Group ring definition

I don't understand the definition of a group ring. In "An Introduction to Group Rings" by Polcino and Sehgal (page 129), an element of the group ring $RG$ is a linear combination of elements from $G$ ...
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1answer
51 views

Wedderburn decomposition of $F(Z_2\times Z_2).$

Let $F$ be any finite field of characteristic different from $2$. I have to prove $F(Z_2\times Z_2)\cong F\oplus F\oplus F \oplus F.$ I know that $F(Z_2\times Z_2)\cong F(Z_2)\otimes_F F(Z_2)$ and $F(...
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40 views

Group rings decomposition

Let $G$ be a group with $|G| = 8$. By the Artin-Wedderburn Theorem, $\mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a ...
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38 views

Exponent of $1+J(GF),$ $J$ stands for Jacobson radical.

Consider the group algebra $F[G]$, where $F$ is a finite field of characteristic $2$ and $G=\operatorname{SL}(2,3)$ i.e. group of $2\times2$- matrices over the integers modulo $3$. After some ...
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30 views

Powers of ideals in group ring $\mathbb{Z}_nS_m$

Let the group ring $\mathbb{Z}_2S_3,$ where $S_3$ is the permutation group on $3$ elements, with presentation $S_3 = \{1, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau \} = \langle \sigma, \tau | \...
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2answers
54 views

How orbits of $G$-sets and these characters are related

I've been learning about induced representations recently and I've come across something which I'm very confused about; For any $G$-set $X$, the number of orbits is equal to $(1_G, \chi_{\mathbb{C}[...
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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 $...
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51 views

Onto Algebra homomorphism between group rings.

I have to determine onto $F$-Algebra map from group algebra $FS_5$ to $M_4(F)$ where $F$ is any finite field of characteristic $2$ and $S_5$ is symmetric group of degree $5$ generated by $a=(1,2,3,4,...
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0answers
50 views

Wedderburn Decomposition by using Clifford theorem.

Let $H$ be a normal subgroup of $G$ and we know Wedderburn decomposition of semi simple algebra $FH$ over a finite field $F$ as $$FH=F\oplus M_{3}(F)\oplus M_{3}(F)\oplus M_{4}(F)\oplus M_{5}(F).$$ ...
2
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1answer
44 views

Dimension of $J(F(GL_2(3)))$, where $J$ stands for Jacobson radical.

I want to find dimension of $J(F(GL_2(3)))$, where $J$ stand for Jacobson radical, for the group algebra $F(GL_2(3))$ of general linear group of two by two matrices over the field $\mathbb{Z}_3,$ and ...
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1answer
21 views

How to do congruence-class arithmetic?

When working through this question: Write out the addition and multiplication tables for the congruence-class ring $F[x]/(p(x))$. $F=\mathbb{Z_2}$; $p(x)=x^{3}+x+1$. [Question #1 in section 5.2: ...
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75 views

Generators of $1+\Delta (G)$, where $\Delta(G)$ is augmentation ideal of group ring $FG.$

Let $FG$ be a finite group ring of a finite non abelian $p$-group $G$ over finite field $F.$ It is well known that augmentation ideal $\Delta(G)=J(FG)$ has basis as the set $\{g-1:g\in G, g\ne 1\}$,...
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1answer
44 views

Group ring isomorphic to matrix ring

I want to figure out whether $R$ is isomorphic to $S$, where $R = \mathbb{R}[G]$, where $G = \mathbb{Z}/2 \times \mathbb{Z}/2$, and $S = M_4(\mathbb{R})$. It seems that they might not be isomorphic, ...
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0answers
26 views

Showing that three rings are isomorphic

Let $C_3 = \langle a | a^3 = e \rangle$ and let $R=(\mathbb{Z}/2)[C_3]$ be the group ring of $C_3$ with $\mathbb{Z}/2$ coefficients. Let $S = (\mathbb{Z}/2)[y]/(y^3-[1])$ and let $T = \mathbb{Z}[x]/(...
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$G$ finite group and $\mathbb{C}[G]$ its group ring, show: $K_i = \sum_{a_k\in cl_G(a_i)} a_k $, then $Z(\mathbb{C}[G])=span_\mathbb{C}(K_1,…,K_n)$ [duplicate]

Given a finite group $G = \{a_1 ,a_2,...,a_n\}$ and $\mathbb{C}[G]$ the respective group ring, $\mathbb{C}[G] = \{\sum z_ia_i : z_i \in\mathbb{C} , a_i\in G\}$. Defining $K_i := \sum_{a_k\in cl_G(a_i)...
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1answer
73 views

Finding idempotents in group algebra over $A_n$

Let $G=A_4$ be the alternating group on 4 letters, and let $R = \mathbb{C}[G]$. Then $$\mathbb{C}[G] = U\oplus U' \oplus U'' \oplus V^{\oplus 3},$$ where $U,U',U''$ are the three 1-dimensional ...
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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 ...
2
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1answer
23 views

Analogue for Group Rings for multiplicative sets

Suppose we have a ring $R$ and a multiplicative set $S$. Can we define $S$ rings analogous to Group Rings? I think we'll have some problems if inverses do not exist in $S$, since when multiplying ...
9
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2answers
321 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=...
4
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2answers
98 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 ...
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0answers
12 views

Reference requested: injective modules and subgroups

Let $G$ be a group, and let $H$ be a subgroup of $G$. Suppose that $E$ is an injective module over $\mathbb{Z}G$. Then I think that, as a consequence of Baer's criterion, $E$ is also an injective ...
9
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2answers
124 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 ...
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1answer
39 views

The elements of the group ring $R = \mathbb{Z}_4\mathbb{C}_2$

Let $\mathbb{C}_2$ denote the group of order $2$ and let $\mathbb{Z}_4$ denote the ring of integers modulo $4$. The group ring $R = \mathbb{Z}_4\mathbb{C}_2$ is commutative. My problem is how to ...
5
votes
1answer
168 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 ...
7
votes
2answers
195 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 ...
0
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0answers
52 views

Classification of homomorphism as abelian group

Consider $f:\mathbb{Z}^3 \rightarrow \mathbb{Z}^4$ $$f(a, b, c) = (a+2b+8c, 2a-2b+4c, -2b+12c, 2a -4b + 4c)$$ Describe the image of this homomorphism as an abstract abelian group. Describe the ...
5
votes
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 $U:\Bbbk G$-$\mathsf{Mod}\to \Bbbk$-$\...
2
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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 ...
3
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1answer
58 views

Decomposition of $\mathbb{C}[G]$ / Orthogonality relations

I am currently working on Jean-Pierre Serre's "Linear representations" german translation in chapter 6 and I do not understand the last part of the proof of the following theorem. a) Let $\rho_i : G \...
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0answers
63 views

Homology of group rings

Let $\mathbb Z G$ be the group ring of $G$. Denote by $\mathbb Z G ^{gp}$ the universal enveloping group of the monoid $(\mathbb Z G,*)$, i.e. the fundamental group of the classyfiyng space of $(\...