# 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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### Can the integral group ring construction be extended to groupoids in such a way that it provides a functor from Groupoids to Rings?

If $G$ is a group, and $R$ is a ring, one can construct the group ring $R[G]$, which is a free $R$-module with basis $G$, and with multiplication coming from the original multiplication of $G$. http:/...
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### Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
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### 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 ...
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### Is the Transpose of a Representation an Equivalent Representation?

Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two ...
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### Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
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### 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 ...
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### 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 ...
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### Modules over a group algebra

Sorry for the bad quality picture. The minuses should of course be equals. For the first, I'm thinking Schurs Lemma will be involved. For ci) I need to show it's a subspace and is closaed under the ...
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### Elements of Group Rings, Are they Reducible?

In general, elements of group rings are written as sums of the group elements multiplied by scalars from the ring, correct? What is the utility of that, if we don't know how to reduce the elements of ...
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### Showing the augmentation ideal of $\mathbb{F}[G]$ is a maximal ideal

Given a basis of $\Bbb R^n,\ G:=\{e_0,...,e_{n-1}\}$, we define multiplication on the elements of the basis by $e_i\cdot e_j=e_{i+j}$ (where $i+j$ is calculated modulo $n$). For a field $\Bbb F$ we ...
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### 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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### Isomorphism of Tensor Product over a Group Ring.

Let $\mathbb{Q}$ be the rationals and $\mathbb{Z}$ integers. Let further $p$ be prime and $t\in \mathbb{Z}$ such that $p \mid t$. Then $\mathbb{Z}_{(p)}$ is the local ring. Let $G < H$ be groups, ...
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### Calculation on Infinite Groups and Group Rings.

Let $G$ be an infinite group and $x \in R[G]$ the group ring of $G$ over $R$. If $gx-x=0$ $\forall g \in G, g\neq 1$, then follows $x=0$. I need a nice proof. I already have one dealing with the '...
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### Zero divisor conjecture

Let $K$ be a field and $G$ be a group. Then $K[G]$ is a domain iff $G$ is torsion-free. I know that "$\Leftarrow$" is conjectured to be always true. But what about the other direction?
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### Finite $\Delta$-module of $p$-power order

I have a question concerning lemma, that I want to prove: Let $p$ be a prime and $\Delta$ be a finite group of order prime to $p$. Let $M$ be a finite $\Delta$-module of order a power of $p$. Then ...
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### 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 ...
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### How to turn a group ring $R(G)$ into a ring?

Let $R(G)$ be a given abelian group ring. Any abelian group ring is isomorphic to an abelian ring. I know how to express (isomorphism) some group rings as a ring. But I wonder if there is a general ...
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### Quotient of ideal in group ring is isomorphic to abelianization [duplicate]

Let $G$ be a group and $\mathbb Z G$ the group ring over the integers. Let $I$ be the ideal of elements $\sum_{g\in G} n_g g$ with $\sum_{g\in G} n_g = 0$. I am trying to prove that $I/I^2$ is ...
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### $G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]

Possible Duplicate: Isomorphism between $I_G/I_G^2$ and $G/G'$ Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal of the integral group ring $\mathbb{Z}[G]$. I'm ...
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### 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 ...
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### How to calculate inverses in the group algebra

Is there an algorithm to calculate the inverse of an element in the group algebra? For example, does the element $(1 2 3) + 2 . (1 2)(3 4)$ in the group algebra $\mathbb{C} S_{4}$ have an inverse, ...
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### 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{...
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