# Tagged Questions

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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### 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 ...
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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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### Units in finite polynomial rings

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where ...
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### Conjecture about some group ring representations.

In this link : http://bandtechnology.com/PolySigned/ A set of numbers is described : $P(N)$. $P(3),P(4),P(5),...$ are all (algebraicly closed) group rings. Identify $PN$ with ...
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### Homomorphism involving a basis for $\Bbb R^n$

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

Let $\mathbb{Z}$ be the integers, $p$ prime and $G$ any torsionfree group. Then I denote the groupring of $\mathbb{Z}$ over $G$ with $\mathbb{Z}[G]$. I am looking for am isomorphism: \begin{align} ...
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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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### When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then we can form the group ring $R(G)$. Note that if $g^{n+1} = e$ then ...
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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 $\mathbb{Z}[G]$. ...
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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]: ...
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### Augmentation ideal of the group ring

Let $G$ be a group and $I_G$ be the augmentation ideal of the group ring $\mathbb{Z}G$, i.e. $I_G$ consists of formal linear combinations $\sum n_i g_i$ ($n_i\in\mathbb{Z}$, $g_i\in G$) such that ...
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 ...
### Isomorphism between $I_G/I_G^2$ and $G/G'$
Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it. $G$ is ...