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

I tried the following: ...
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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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### 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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### 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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### 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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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 $K_i := \sum_{a_k\in cl_G(a_i)... 1answer 73 views ### Finding idempotents in group algebra over$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 ... 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 ... 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 ... 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 $$x=... 2answers 98 views ### Question on Group ring \mathbb{Q}G where G is a finite group Let G=\{e=g_1,g_2,.....g_n\} be a finite group of order n and let \mathbb{Q}G be the group ring. Let \sigma=\sum_1^ng_i. Prove that \sigma^2=n\sigma and deduce that \mathbb{Q}G has a ... 0answers 12 views ### Reference requested: injective modules and subgroups Let G be a group, and let H be a subgroup of G. Suppose that E is an injective module over \mathbb{Z}G. Then I think that, as a consequence of Baer's criterion, E is also an injective ... 2answers 124 views ### Reference for Kaplansky's proof that in \mathbb{C}[G], ab=1 implies ba=1 Here on the Wikipedia page for Group Rings, talking about group rings over infinite groups, The case [of a group ring R[G] where G is an infinite group] where R is the field of complex ... 1answer 39 views ### The elements of the group ring R = \mathbb{Z}_4\mathbb{C}_2 Let \mathbb{C}_2 denote the group of order 2 and let \mathbb{Z}_4 denote the ring of integers modulo 4. The group ring R = \mathbb{Z}_4\mathbb{C}_2 is commutative. My problem is how to ... 1answer 168 views ### When is a Hopf Algebra isomorphic to a group ring k[G]? Let H be a Hopf algebra over a field k. What are some nice conditions for when H is isomorphic to k[G] for a finite group G? The co-multiplication structure on the group algebra k[G] is ... 2answers 195 views ### Isomorphism of group rings over finite cyclic groups Let R be a commutative ring with unity. For any group G, let the group ring be denoted by R[G]. If R[\mathbb Z/(n) ] \cong R [\mathbb Z / (m)] as rings , then is it true that m=n ? If that ... 0answers 52 views ### Classification of homomorphism as abelian group Consider f:\mathbb{Z}^3 \rightarrow \mathbb{Z}^4$$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 ... 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$U:\Bbbk G$-$\mathsf{Mod}\to \Bbbk$-$\...
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 ...