# 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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### Units of group algebra of dihedral group

Question: Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite ...
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### Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?

Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
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### Units in the group ring over fours group after Gardam

Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$. This conjecture asserted that if $\Gamma$ is a ...
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### Integral monoid rings and Ore conditions

Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$. I have two, ...
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### Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
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### How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
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### Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$

Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...
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### Flat augmentation ideal of a group-ring

If $G$ is a group and $I$ the augmentation ideal $I=Ker(\mathbb{Z}G\rightarrow \mathbb{Z})$ suppose that: $I$ is a flat (right) $\mathbb{Z}G$-module. $I$ is a finitely generated (right) $\mathbb{Z}G$...
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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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### Any f.p. faithful simple module over a primitive group ring?

Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$. There ...
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### 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 $\varphi(n)=n$? ...
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### 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 ...
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### 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 $\mathbb{K}[H]$ ...
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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}$$(\... 2answers 661 views ### Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial? Let G be a (possibly infinite) group. Let \mathbb{Z}[G] be its integral group ring and let P be a finitely generated projective module over \mathbb{Z}[G]. Suppose that the coinvariants of P ... 1answer 219 views ### Why do we not lose any generality by proving it only for finitely generated groups [closed] In the proof of following theorem, in a paper by Farkas- Here \Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \} and U_1(\mathbb{Z}G) is the set of normalized units of the integral group ring \... 1answer 210 views ### Units in a finite semisimple group algebra Let G be a finite group and k a finite field, with the characteristic of k not dividing the order of G. Then kG is a finite semisimple group algebra with the interesting property that an ... 3answers 1k views ### A semisimple group ring Let n \in \mathbb{N}, p a prime number, and G a finite group of order coprime to p. Let R = \mathbb{Z} /p^n \mathbb{Z} be the ring of integers mod p^n. Must R[G] be semisimple? As noted ... 1answer 2k views ### Does GL_n(Z) have a noetherian group ring? Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like GL_n(Z) been considered in the literature? Motivation: a recent trend has been to study "... 2answers 723 views ### Kaplansky's idempotent conjecture for Thompson's group F Let K be a field and G be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring K[G] does not contain any non-trivial idempotent, i.e. if x^2=x then x=0 or x=1.... 1answer 203 views ### Find a special element in group algebra Let$$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 ... 0answers 147 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 ... 1answer 202 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? 0answers 260 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}...
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....
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 ...