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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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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Homomorphism of group rings.

This is an example from Hungerford's textbook ''Algebra''. Let G and H be multiplicative groups and $f:G \rightarrow H$ a homomorphism of groups. Let R be a ring and define a map on the group rings $...
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1answer
100 views

Showing an isomorphism of $R[G]$-modules using the regular representation

Let $R$ be a commutative ring, and $G$ a finite group. Now suppose $M,N$ are finitely generated $R[G]$-modules such that $M\cong_R N$ (let's say they have $R$-rank$=n$). To show $M\cong_{R[G]}N$, is ...
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104 views

Function such that f(x) = (f(x))^2

Are there any functions of that type? IE. idempotent but not on composition, through function multiplication instead. Sorry, I forgot to mention this, but $f(x)\not\equiv 1$ and $f(x) \not\equiv 0.$
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Introduction to group rings (reference request)

I'm looking for a thorough introduction to group rings, specifically the simple case of group rings over the integers where the group is abelian and finitely generated. I realise that these are ...
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79 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 ...
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371 views

Doubt about the definition of $G$-coinvariant

I am learning some group cohomology from Romyar Sharifi's online notes. (Everything I have written here can be found on page 5-6) Let $G$ be a finite multiplicative group. Consider the group ring $\...
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1answer
209 views

If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring.

If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring. My failed attempt: Since $G$ is finite and $p$ divisdes $|G|$ ...
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474 views

Isomorphism Of Quotient rings [closed]

I have two questions that I need help about. 1) Let $\varphi :R\to S$ be an onto homomorphism, and $I$ ideal in $S.$ Prove that $R/\varphi^{-1} (I)$ isomorphic to $S/I.$ 2) Let $I$ be an ideal in $...
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94 views

General form of Maschke's Theorem for $R[G]$

I've come across some reference to a form of Mashke's theorem for rings; namely, $R[G]$ is semisimple if $R$ is a commutative ring (not necessarily a field) and $G$ is a finite group such that $|G|$ ...
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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 ...
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133 views

Computing the Jacobson radical of $F[G]$ with $char(F)=p$ and $G$ finite with a normal $p$-Sylow subgroup

Let $G$ be a finite group and $F$ be a field of prime characteristic $p$. Suppose further that $G$ has a normal $p$-Sylow subgroup $N$. Question: Is it true that the Jacobson radical of the ...
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1answer
50 views

${I}=\Delta(G,G')$ is the smallest ideal of the group ring $\mathbb{Z}{G} $ such that $\mathbb{Z}{G}/{I}$ is a commutative ring

How can I prove that if ${G}$ is a group then ${I}=\Delta(G,G')$ is the smallest ideal of the group ring $\mathbb{Z}{G} $ such that $\mathbb{Z}{G}/{I}$ is a commutative ring? It is an ideal is clear, ...
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70 views

Some results about Group Rings/Algebras

I'm trying to build up my intuition on group algebras, $k[G]$ where $k$ is a field. Here are some things I'd like to know about: If $H \leq G$ then is $k[H]$ a subalgebra of $k[G]$? If $G_1, ...
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235 views

Canonical homomorphism from a group ring to its subring.

If $R(G)$ is a group ring constructed from a group $G$ and a ring $R$, and $H\leq R(G)$ is a subring, then must there be a surjective homomorphism from $R(G)$ to $H$ in general? What if $G$ is a ...
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398 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 ...
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53 views

What is $\mathbb{Q}[\mathbf{Z}/n\mathbf{Z}]$?

I was thinking, you can send $\sum f(g)g\in\mathbb{Q}[\mathbb{Z}/n\mathbb{Z}]$ to $(f(g))_{g\in G}\in \mathbb{Q}\times\cdots\times\mathbb{Q}$. This is surjective and has zero kernel, so this map ...
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1answer
802 views

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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44 views

$(\mathbb{F}_p\times\mathbb{F}_p)[F_n]\cong\mathbb{F}_p[F_n]\times\mathbb{F}_p[F_n]$ where $p$ is prime and $F_n$ is the free group of rank $n$.

Let $p$ be an odd prime and $F_n$ the free group of rank $n$. I want to show the following isomorphism of group rings: $(\mathbb{F}_p\times\mathbb{F}_p)[F_n]\cong\mathbb{F}_p[F_n]\times\mathbb{F}_p[...
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Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I ...
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1answer
63 views

Looking for a good name for this “Quantisation Regime”

This is an attempt to salvage the wreckage of my hope to motivate (finite) quantum groups a lá this question. Let $\{S_i\}_{i=0}^n$ be a family of finite sets and let $\varphi$ be a map $$\varphi:...
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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 $...
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137 views

A question about Group Rings

Let $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integers module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, u_1^p=1,\,\,u_{...
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1answer
150 views

States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
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1answer
63 views

Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?

Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...
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92 views

Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
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1answer
152 views

Natural action of $\mathbb{Z}G$ on $\mathbb{Z}$?

I'm studying projective modules and I'm having problem coming up with (or understanding) examples of non-free projective modules. I got that when a ring is a direct sum $R = A \oplus B$, both $A$ and $...
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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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1answer
24 views

Show that $[u,g]\in G'$ where $u\in N_U(G)$

Let $G=\langle H,g\rangle$ where H is an abelian subgroup of index $2$. Let $\Bbb{Z}G$ be the group ring and $u$ be a unit of $\Bbb{Z}G$ which normalizes $G$. Then we can write $u$ as $\alpha_1+\...
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1answer
747 views

A group-ring is commutative if and only if that group is abelian

Problem says: Let $R$ be a nontrivial commmutative ring and $G$ a group. Prove that $R[G]$ is commutative if and only if $G$ is abelian. I solved ($\Rightarrow $) direction as follow: Suppose ...
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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 ...
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911 views

Compute the Jacobson radical of the group ring $\mathbb{F}_2S_3$.

Compute the Jacobson radical and the maximal semisimple quotient of the group ring $\mathbb{F}_2S_3$ of the symmetric group on three letters over the field with two elements, and compute the same ...
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1answer
33 views

Which element of $\mathbb{Q}C_5$ will be mapped onto $(0,4)\in \mathbb{Q} \oplus \mathbb{Q}(\zeta)$

I know that the rational group algebra of $C_5 = \langle g : g^5=1\rangle$ is $\mathbb{Q}C_5 \cong \mathbb{Q} \oplus \mathbb{Q}(\zeta)$ where $\zeta$ is a root of $x^4+x^3+x^2+x+1$. But I was ...
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1answer
64 views

prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$

I want to prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$, where $C_2=<x| x^2=1>$, where $\mathbb{Z}(G\times C_2)$ and $(\mathbb{Z}G)C_2$ are integral group rings and I am looking for ring ...
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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}(\...
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1answer
108 views

Morita equivalence between $\mathbb{C}[G]$ and $\mathbb{C}[H]$?

What we can say about two groups G and H when their group rings, $\mathbb{C}[G]$ and $\mathbb{C}[H]$, are Morita equivelent?
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Structure of $\mathbb{Q}S_3$

I have an exercise to show that $\mathbb{Q}S_3 \cong \mathbb{Q} \oplus\ \mathbb{Q}\ \oplus\ M_2(\mathbb{Q}) $ , where $M_2(\mathbb{Q})$ is ring of $2$ by $2$ rational matrices and $\mathbb{Q}S_3$ is ...
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1answer
39 views

If a group algebra acts regularly on a module, can this module be identified as a left ideal?

To be more specific, I am looking at $F_2[D_p]$, where $D_p$ is the dihedral group of order $2p$. If this group acts regularly on the basis of a vector space $F_2^{2p}$, and there is a subspace of $...
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1answer
92 views

Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID?

Is $\mathbb{Z}_{p}[G]$ a PID, where $G=(\mathbb{Z}_{p},+)$ is the additive group of the $p$-adics $\mathbb{Z}_{p}$? I am studying a paper where the authors implicitly use that claim, but it is ...
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1answer
57 views

Is $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID?

As the title suggests, I'm interested whether $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID or not. Assume $p$ is prime. My feeling is that it is a PID, since $\mathbb{Z}/(p)$ is cyclic an morally if an ideal ...
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1answer
121 views

Prove that $u$ is a trivial unit

Let $G$ be a finite group. I want to prove that if $u\in \mathbb{Z}G$ be a torsion unit (i.e. for some $n$, $u^n=1$) of integral group ring $\mathbb{Z}G$ such that $u$ normalizes $G$, then $u$ is a ...
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1answer
50 views

Constructing Factor Ring $\mathbb C / I$ where $I=_{\mathbb C[X]}\langle X^2+1 \rangle$

I'm trying to get some intuition behind the Factor Ring $\mathbb C[X] / I$ where $I=_{\mathbb C[X]}\langle X^2+1 \rangle$. 1)What does it look like in the set notation? 2)Is the equivalence ...
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45 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 ...
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1answer
112 views

Tensoring Groupring with field of fractions

Is it true that $\mathbb Z[G]\otimes_\mathbb Z \mathbb Q$ is isomorphic to $\mathbb Q[G]$ as a $\mathbb Q$-algebra? Context: If I have the Jacobian of a smooth curve $X$, call it $J_X$, then I've ...
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1answer
219 views

Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace

The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of ...
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1answer
280 views

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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1answer
71 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
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2answers
78 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} $...
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2answers
120 views

If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$

Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism $$ \varphi(1) = (1,0), \...
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1answer
29 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 ...