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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288 views

Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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290 views

The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
6
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196 views

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{...
5
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178 views

No onto map between group algebras $FS_5$ onto $M_6(F)$.

I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $...
5
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76 views

When is $R^{op}\cong R$?

Suppose $R$ is a noncommutative ring. When could I reasonably expect $R^{op}\cong R$? For instance I know group rings have a natural involution, i.e. if $SG$ is the group ring in question then $r\...
5
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1answer
83 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
5
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202 views

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 ...
4
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41 views

$\mathbb{Z}_p[\mathbb{Z}/p^{n}\mathbb{Z}]\cong \mathbb{Z}_p[T]/\left((T+1)^{p^n}-1\right)$ as topological rings?

Consider the group-ring $\mathbb{Z}_p[\mathbb{Z}/p^{n}\mathbb{Z}]$ with the product topology, and the quotient ring $\mathbb{Z}_p[T]/((1+T)^{p^n}-1)$ with the quotient topology, ($\mathbb{Z}_p[T]$ has ...
4
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140 views

Why are integral group rings so important in topology?

Having worked on group rings $\mathbb{Z}[G]$ for the last year, I am beginning to feel quite comfortable with them. I also know of a few applications to topology. However, these are individual uses ...
4
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200 views

A question regarding isomorphism of group rings

For a ring with unity $R$ and a group $G$ let $R[G]$ denote the group ring. Now let $R$ be a commutative Noetherian ring with unity such that $R[\mathbb Z_m] \cong R[\mathbb Z_n]$ (isomorphic as rings)...
4
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77 views

When does an integral group ring have finite global dimension?

Let $G$ be a finite group and $R=\mathbb{Z}[G]$ the integral group ring. If $G$ is such that $R$ is Noetherian (so $G$ polycyclic-by-finite) when does $R$ have finite global dimension? Another way of ...
4
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269 views

Generation of left ideals in group rings

This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated. Suppose that we are given an (infinite) ...
3
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108 views

Group algebra functor preserves colimits

Consider a commutative unital ring $R$ and the group algebra functor $$R[-]:\bf{Grp}\rightarrow {Alg}_R$$ which has the group of units functor $$(-)^\ast:\bf{Alg}_R\rightarrow \bf{Grp}$$ as right-...
3
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126 views

Fraction field of group ring of field over torsion free abelian group

Let $G$ be a torsion-free abelian group. If $k$ is a field, it is known that $k[G]$ is an integral domain. Let $k(G)=\operatorname{Frac} k[G]$. If $G,H$ are torsion free abelian groups such that $k(G) ...
3
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48 views

Set of functions from a finite field to the integers

Has the set of functions $\mathbb{F}_q \to \mathbb{Z}$ endowed with pointwise addition and additive/multiplicative convolution been studied? Does anyone know a reference or keywords to search for? ...
2
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157 views

Representation of group ring $R[G]$

Been studying Group Rings and their applications to the point that I can represent an element $a \in RG$ as a $n\text{x}n$ matrix where $n=\vert G \vert$. Besides being succesful, I've found others ...
2
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90 views

Trivial units of commutative group ring

I want to proof the following equivalence: All units of a commutative group ring $\mathbb{Z}G$ are trivial $\Leftrightarrow$ for every $x \in G$ and every natural number $j$, relatively prime to $|G|$,...
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83 views

Units in a group ring $\mathbb{Z}[G]$

This is a homework question: For a group G, let $g\in G$ have finite order, such that $\langle g\rangle$ is not a normal subgroup of $G$. Then $\mathbb Z[G]$ has a unit other than $\pm h$ with $h\...
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28 views

Express $[RG,RG]$ as $R-$ linear span of $[g,h]$

Let $R$ be a ring and $G$ be a group and $RG$ be the group ring. Denote by $[R,R]$, the additive subgroup generated by all lie products $[x,y]=xy-yx , \forall\ x,y\in R$. Then how is this that $[\...
2
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42 views

Index of $\mathbb{Z}G$ in the maximal Order for cyclic $G$.

I am stuck at Question and I hope someone can help me. So, let $G$ be a finite cyclic group of order $n$. I am able to proof the following isomorphisms for the group algebra $\mathbb{Q}G$: $$\mathbb{...
2
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209 views

The augmentation ideal of $\mathbb{Z}G$

Let $G$ be a cyclic group of order $p$ and let $IG$ denote the augmentation ideal of the group ring $\mathbb{Z}G$. I need to find $H^1(G,IG)$. Since $$0 \rightarrow IG \rightarrow \mathbb{Z}G \...
2
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392 views

Best book to understand representation theory.

I have tried to read representation and character theory from a few books but none of them was working for me, like Lieback and Serre. I want to understand representation and character to use them and ...
2
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58 views

What's the definition of $x^g$?

My question is really simple. I'm studying a book of group rings which doesn't define what is $x^g$, where $x,g$ are elements of the group $G$: Thanks
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42 views

Is this problem still open:If G is a torsion free group and F is a field then group ring F[G] is an intergral domain.?

I know this question has answer for when G is infinite cyclic group group.Does there is a general proof? Could anyone give me some references...
2
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142 views

An invariant submodule of a projective module

Let $R$ be a commutative unital ring and $S$ be a normal subgroup of a finite group $G$. If $P$ is a finitely generated projective left $R[G]$-module then is the submodule of $S$-invariants $$P^S:= \{...
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45 views

What is the differential $d^0(f) : C^0(G, A) \to C^1(G, A)$ knowing the general formula?

Romyar Sharifi's Lecture Notes It's from the first differential formula appearing on page 7 and that formula is: $$ d^i : C^i(G, A) \to C^{i+1}(G, A), \\ d^i(f)(g_0, \dots, g_i) = g_0 f(g_1, \dots, ...
1
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40 views

Group ring $R[G]$ semisimple if and only if $J = J^2$?

Let $G$ be an abelian group and let $R$ be a commutative ring and consider the group ring $R[G]$ of finite formal linear combinations of elements of $G$ with coefficients in $R$. Let $J = (1 - g ~|~ g ...
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38 views

Exponent of $1+J(GF),$ $J$ stands for Jacobson radical.

Consider the group algebra $F[G]$, where $F$ is a finite field of characteristic $2$ and $G=\operatorname{SL}(2,3)$ i.e. group of $2\times2$- matrices over the integers modulo $3$. After some ...
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51 views

Onto Algebra homomorphism between group rings.

I have to determine onto $F$-Algebra map from group algebra $FS_5$ to $M_4(F)$ where $F$ is any finite field of characteristic $2$ and $S_5$ is symmetric group of degree $5$ generated by $a=(1,2,3,4,...
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50 views

Wedderburn Decomposition by using Clifford theorem.

Let $H$ be a normal subgroup of $G$ and we know Wedderburn decomposition of semi simple algebra $FH$ over a finite field $F$ as $$FH=F\oplus M_{3}(F)\oplus M_{3}(F)\oplus M_{4}(F)\oplus M_{5}(F).$$ ...
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75 views

Generators of $1+\Delta (G)$, where $\Delta(G)$ is augmentation ideal of group ring $FG.$

Let $FG$ be a finite group ring of a finite non abelian $p$-group $G$ over finite field $F.$ It is well known that augmentation ideal $\Delta(G)=J(FG)$ has basis as the set $\{g-1:g\in G, g\ne 1\}$,...
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65 views

Homology of group rings

Let $\mathbb Z G$ be the group ring of $G$. Denote by $\mathbb Z G ^{gp}$ the universal enveloping group of the monoid $(\mathbb Z G,*)$, i.e. the fundamental group of the classyfiyng space of $(\...
1
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1answer
38 views

The isomorphic between rings

Suppose $\Gamma$ is a finite group ,$R$ is a commutative ring with $1$. Then the set of maps between $\Gamma$ and $R$ become a commutative ring . The zero element is the zero map ,the identity is the ...
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40 views

A $\mathbb Z_p$-algebra homomorphism from $\mathbb Z_p[[T]]$ determined by its value on $1+T$ (?)

Let $f$ and $g$ be $\mathbb Z_p$-algebra homomorphisms from $\mathbb Z_P[[T]]$ to $\varprojlim\limits_{n} \mathbb Z_p[\Gamma/\Gamma^{p^n}]$, where $\Gamma$ is the abelian group $\mathbb Z_p$ written ...
1
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0answers
39 views

Show that any $f \in \mathbb{Q}(\langle x \rangle)$ is associate to any $\overline{f} \in \mathbb{Q}[x]$

Let $G = \langle x \rangle$ be the infinite cyclic group generated by an element $x$. The group ring $R = \mathbb{Q}(G)$ consists of finite sums of the form $$ r_{-m}x^{-m}+r_{-m+1}x^{-m+1}+\cdots + ...
1
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2answers
51 views

Is there an 'easy' way to calculate $K_0(\mathbb{Z}[C_p])$?

For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal ...
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0answers
33 views

Decomposing $\mathbb{F}_p[G]$ ($G$ finite) into products of matrix rings over fields

I have recently begun learning about group algebras over finite fields but am still a little uncertain about these guys. So I was looking for some clarification and verification. Consider the ...
1
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0answers
97 views

submodules of a group algebra

Estoy viendo el álgebra de grupo $KG$ donde $K$ un cuerpo y $G$ un grupo finito. Definimos $KG=\left\{{\sum_{g\in{G}}^{}k_gg : k_g\in{K}, g\in{G}}\right\}$ y la llamamos álgebra de grupo. $KG$ es un $...
1
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0answers
44 views

Properties of Linear Transforms

Given a linear function $T: \Bbb R \rightarrow \Bbb R$ and $T(x+y) = T(x) + T(y)$, do we have that $T(x) = s(x)$ for some $s \in \Bbb R$ I'm doing an exam review for my analysis class but the ...
1
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0answers
49 views

Isomorphism map from $\Bbb{Q}(C_2\times C_2) $ to $\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} $

I know how to find structure of Rational group algebras of finite cyclic groups such as for cyclic group $C_6=\langle a \rangle$ we have $\Bbb{Q}C_6 \cong \Bbb{Q} \oplus \Bbb{Q}\ \oplus\ \Bbb{Q}(\...
1
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0answers
130 views

Jacobson radical of integral group ring ZS3

I want to know whether Jacobson radical of integral group ring ZS3 (S3 is symmetric group) is known or not. Please help me. Tell me about any reference. Also if anyone know about the maximal ideals ...
1
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1answer
28 views

Equivalent condition for normalizer problem

Let $\cal{U}$ be the unit group of group ring $\Bbb{Z}G$ then the Normalizer Problem (NP) states that $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}=\cal{Z(U)}$. Now why (NP) is equivalent to saying that ...
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0answers
33 views

What is the augmentation ideal of $\Bbb{Z}S_3$

I know that $\Delta_{\Bbb{Z}}(G)$ is the $\Bbb{Z}-$ module generated by elements of form $\{g-1\ \forall\ g\in G\}$. But how do we find them or what it looks like. I was thinking about finding aug ...
1
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0answers
97 views

Calculation in a Group Ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
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0answers
28 views

how to prove intersection of normal subgroups controls the ideal

I have to prove the following: Let $I$ be an ideal of $K[G]$ ($K$ is a field, $G$ is a multiplicative group, $K[G]$ is a group ring) and let $H_1,\ldots,H_n$ be normal subgroups of $G$, each ...
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0answers
88 views

Ph.D thesis by Whitcomb

Does somebody has a link to Ph.D thesis by Whitcomb titled "The group ring problem" University of Chicago 1968. It was referred in a paper and I have some things to look up in that. I could not find ...
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0answers
66 views

$\mathbb{Z}G\otimes_{\mathbb{Z}N} \mathcal{N}= \mathcal{N}^G$ is a non-trivial idempotent ideal in $\mathbb{Z}G$

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 $\...
1
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0answers
23 views

Unit group of $\mathbb{Z}C_m$

By Higman's theorem we know that if $G$ is abelian and finite then $\mathbb{Z}G$ has only trivial units iff G is has exponent $1,2,3,4, \text{or}\ 6$ But what if $G=C_{m}$ where $m \ge 7$. In this ...
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0answers
22 views

If $U(\mathbb{Z}G)$ is nilpotent/ FC group than $TU(\mathbb{Z}G)$ is a subgroup.

I am doing a short paper by Miles and Parmenter "GROUP RINGS WHOSE UNITS FORM A NILPOTENT OR FC GROUP" and the main theorem in that is the following- Now for (i) or (ii) implies (iii) he writes that ...
1
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0answers
19 views

Does $U=U_1(\mathbb{Z}G)$ normalize $G$?

Let $G$ is an arbitrary group and and $U=U_1(\mathbb{Z}G)$ is the set of normalized units of $ZG$ i.e. $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. set of all ...