# Questions tagged [group-rings]

A group ring $R[G]$ is a ring constructed in a natural way from a ring $R$ and a group $G$.

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votes

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253 views

### Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$?
...

**3**

votes

**0**answers

88 views

### Does this element belong to $\mathbb CG$?

Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...

**5**

votes

**0**answers

151 views

### Are these element in a group algebra of a torsion-free group zero divisors?

Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...

**5**

votes

**0**answers

117 views

### lifting of idempotents in group ring

Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...

**29**

votes

**0**answers

433 views

### Groups whose complex irreducible representations are finite dimensional

By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...

**11**

votes

**1**answer

167 views

### Group rings such that every (countably generated) module has a maximal submodule

Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
...

**7**

votes

**2**answers

215 views

### Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring

Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" component is a ...

**3**

votes

**1**answer

213 views

### Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...

**5**

votes

**0**answers

114 views

### Chains of right annihilators in group rings

See the update below
This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay.
Let $G$ ...

**1**

vote

**0**answers

36 views

### When is genus the same as stable equivalence?

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...

**6**

votes

**1**answer

223 views

### Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$

Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say ...

**2**

votes

**0**answers

72 views

### Improved dimension subgroups

Given a group $G$,
one can define
(see below)
a descending sequence of subgroups
$K^s(G)$, $s=1,2,\dots$,
satisfying
$$
\gamma^s(G)\subseteq K^s(G)\subseteq D^s(G),
$$
where
$\gamma^s(G)$ is ...

**4**

votes

**1**answer

117 views

### Right reversibility of submonoids of nilpotent groups

Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$).
A (cancellative) monoid/semigroup $S$ is right reversible if for ...

**6**

votes

**2**answers

450 views

### In the group ring $\mathbb{Z}_p [G]$, what elements satisfy $(\sum a_g g)^p = \sum a_g g^p$?

Here $\mathbb{Z}_p$ is the ring of integers in $\mathbb{Q}_p$.
Preferably I would want to know this for a general group $G$, but I have been concentrating on the case $G = (\mathbb{Z} / p^n \mathbb{Z}...

**1**

vote

**0**answers

90 views

### Is every stably free module of commutative group ring free?

Is every stably free module of commutative group ring free? if not can you give me an example of commutative group ring with nonfree stably free module.

**5**

votes

**2**answers

151 views

### groupring morphisms and bialgebra

Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...

**1**

vote

**2**answers

301 views

### Jacobson radical of group algebra

For a finite group G and a finite field $\mathbb{F}_p$ of characteristic $p$,
J($\mathbb{F}_{p^k} \otimes_{\mathbb{F}_p }\mathbb{F}_p G ) = J(\mathbb{F}_{p^k}G)$? where $J(\mathbb{F}_{p^k}G)$ is the ...

**5**

votes

**0**answers

154 views

### The augmentation filtration on a group ring

Let $G$ be a group and $\mathbb QG=\{\sum a_ig_i\colon a_i\in\mathbb Q, g_i\in G\}$ its group ring over $\mathbb Q$. It is a Hopf algebra. Let $I=\{\sum a_ig_i\colon\sum a_i=0\}$ be the augmentation ...

**9**

votes

**0**answers

193 views

### Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...

**13**

votes

**2**answers

758 views

### Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture

Let $G_1$ and $G_2$ be the groups with the following presentations:
$$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$
$$G_2=\langle a,b \;|\; ...

**1**

vote

**0**answers

364 views

### Invertible elements in a group algebra

Let $H$ be a torsion-free abelian group and let $\mathbb{K}$ be a field with two elements.
I would like to ask the following question:
Is the group of units of the group algebra $\mathbb{K}[H]$ ...

**1**

vote

**0**answers

155 views

### The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\...

**7**

votes

**2**answers

508 views

### Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?

Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...

**-1**

votes

**1**answer

191 views

### Why do we not lose any generality by proving it only for finitely generated groups [closed]

In the proof of following theorem, in a paper by Farkas-
Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring $\...

**3**

votes

**1**answer

170 views

### Units in a finite semisimple group algebra

Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...

**1**

vote

**2**answers

797 views

### A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted ...

**7**

votes

**2**answers

604 views

### Kaplansky's idempotent conjecture for Thompson's group F

Let $K$ be a field and $G$ be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring $K[G]$ does not contain any non-trivial idempotent, i.e. if $x^2=x$ then $x=0$ or $x=1$....

**0**

votes

**1**answer

195 views

### Find a special element in group algebra

Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ 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 ...

**0**

votes

**0**answers

138 views

### Using extended group rings for combinatorial generating functions

In work of mine recently, I have come to investigate generalised recurrence relations. The generalisation I have in mind is where, instead of natural numbers or integers, the recurrence is over some ...

**2**

votes

**1**answer

192 views

### Description of the units of the group ring Fp[Fp] ?

Is there a good way to see what the units of the group ring $\mathbb{F}_p[\mathbb{F}_p]$ (p is a prime) are?

**3**

votes

**0**answers

255 views

### How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...

**5**

votes

**0**answers

635 views

### Unitary unit conjecture for group rings

The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g....

**53**

votes

**1**answer

5k views

### What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...

**3**

votes

**4**answers

2k views

### When a group ring is a local ring [closed]

Hi there, I'm stuck with my undergraduate thesis on the following proposition:
If $k$ is a field of characteristic $p > 0$ and $G$ is a finite $p$-group, then the group ring $kG$ is local.
In ...

**5**

votes

**1**answer

393 views

### Group ring computation

Let $G$ be a finite abelian group. Is it true that the following element of the group ring ${\mathbb Z}[G]$:
$$
\prod_{g\ne 1}(1-g)
$$
is non-zero?

**5**

votes

**3**answers

2k views

### Do the homological dimension and cohomological dimension for a group agree?

Or equivalently, if $G$ is a group, do the projective and injective dimension of $Z$ (viewed as a $ZG$-module) agree?
Thanks!