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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22 views

Flat extensions of group rings

Let $R$ be a commutative ring, $f:H\to G$ a surjective group homomorphism and consider $RG$ as a $(RG,RH)$-module via $g\cdot h := g\cdot f(h)$ as usual. Now suppose that $RG$ is flat over $H$, ...
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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} \...
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Jacobson radical of integral group ring $\mathbb ZS_3$

I want to know whether Jacobson radical of integral group ring $\mathbb ZS_3$ ($S_3$ is the symmetric group) is known or not. Please help me. Tell me about any reference. Also if anyone knows about ...
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Group ring generated by $\mathbb Z$ and the quaternion group

I would like to calculate general $n$th power of $i+j$ in the group ring. My idea was to find some patterns after calculating some powers of $i+j$, conjecture the general form of $n$th power of it ...
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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=...
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23 views

Cycle detection using group ring membership.

Let $K=F_2$ and $A \in F_2^{n \times n}\quad$ which is the adjacency matrix of an arbitrary graph. We want to determine what permutation cycles are embedded in $A$, for that select $F_2G$ where $\vert ...
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40 views

Group ring of $\mathbb Z\times \mathbb Z$.

We know that the group ring $\mathbb Z[\mathbb Z]$ of $\mathbb Z$ is just the Laurent polynomial ring $\mathbb Z[t^{\pm1}]$. I want to know about the group ring $\mathbb Z[\mathbb Z\times \mathbb Z]$ ...
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1k views

Units of a group ring.

Let $\mathbb{Q}$ be the rationals and $G$ a group. Then we consider the group ring $\mathbb{Q}[G]$. Since the operation on $\mathbb{Q}[G]$ restricted to $G$ is just the group operation, I know that ...
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8 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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Representation of group ring $R[G]$

Been studying Group Rings and their applications to the point that I can represent an element $a \in RG$ as a $n\text{x}n$ matrix where $n=\vert G \vert$. Besides being succesful, I've found others ...
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1answer
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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 $k[G]$-...
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1answer
26 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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1answer
35 views

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

I tried the following: ...
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45 views

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
88 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
21 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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17 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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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
35 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 ...
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1answer
65 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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1answer
52 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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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 ...
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1answer
39 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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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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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 $G$-...
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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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24 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
70 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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43 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....
2
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1answer
53 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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70 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 $G$ ...
2
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1answer
53 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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48 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
57 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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52 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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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 $$FH=F\oplus M_{3}(F)\oplus M_{3}(F)\oplus M_{4}(F)\oplus M_{5}(F).$$ ...
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1answer
22 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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77 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
68 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
56 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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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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19 views

$G$ 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
77 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 ...
2
votes
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 ...
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 ...