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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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 ...
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 ...
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 ...