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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Group ring confusion

This actually causes a lot of confusion. For a finite group $G$ and a commutative unit ring $R$ I’m trying to prove that $h\in R[G]$ defined as $h=\sum\limits_{g\in G}g$ is in the center of $R[G]$. ...
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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-...
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About the matrix representation of group algebra

Consider the group algebra $\mathbb{R}[C_3]$,where $\mathbb{R}$ is real field and $C_3$ is $3$-order cyclic group. It's known $C_3$ can be represented as $\{1,e^{2\pi i/3},e^{4\pi i/3}\}$, I tried to ...
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What breaks if I use a $G$-module instead of a $\mathbb{K}[G]$-module: Induced reps, Frobenius reciprocity?

The Question I use $\mathbb{K}G$-module to denote a $G$-action on a vector space over $\mathbb{K}$ (side question - is this the standard notation?). A $\mathbb{K}[G]$-module differs in that we allow ...
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1answer
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Why does the product in a group ring have finite support?

Let $G$ be a group, $R$ be a ring. One can then define the groupring $RG = \{f: G \to R \mid \sup(g) \ \mathrm{is \ finite}\}$, with pointwise addition and with the following multiplication: If $\...
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Which rings arise as a group ring?

Let $R$ be an arbitrary associative ring with identity. When does there exist a group $G$ and a field $F$ such that $F[G] = R$? Do we obtain more rings as $F[G]$ if we loosen the condition that $F$ ...
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75 views

Morita equivalence between $k$ and $kG$

I would like to show that $k$ is Morita equivalent to $kG$ iff $G$ is the trivial group. clearly, if $G$ is the trivial group, $kG\cong k$ and so $k$ is Morita equivalent to $kG$ However for the ...
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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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62 views

Concerning subrings of a integral group ring

Let $G$ be a group . Does a subring of the integral group ring $\mathbb{Z}[G]$ has the form $\mathbb{Z}[H]$ for a subgroup $H$ of $G$? Thanks in advance.
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131 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 ...
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Thinking about the group algebra $k[G]$ as functions on $G$

Given a group $G$ and field $k$ one can define the group algebra $k[G]$ in two ways: The underlying vector space of $k[G]$ is the free $k$-vector space on $G$, and the multiplication on $k[G]$ is the ...
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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 ...
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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 ...
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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)...
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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\...
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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) ...
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199 views

Torsion free abelian groups $G,H$ such that $k[G] \cong k[H]$ (as rings) for any field $k$

Let $G,H$ be torsion free abelian groups such that $k[G] \cong k[H]$ for any field $k$. Then is it true that $G \cong H$ ? If this is not true, then what if I change the hypothesis to $R[G]\cong R[H]...
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76 views

Ring isomorphism for $k(G \oplus \mathbb Z )$ with $G$ torsion-free and abelian

Let $k$ be a field and $G$ be a torsion-free abelian group. Then $k[G]$ is an integral domain. If we denote its field of fractions by $F = k(G)$, is it true that $k(G \oplus \mathbb Z )\cong F(X)$? ...
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307 views

Simple modules of the group algebra $kC_p$ over different fields $k$

I want to find the simple modules for the group algebra $A=kC_7$, where $k$ is a field and $C_7$ is the cyclic group of order $7$. When $k = \mathbb{C},$ by Maschke's Theorem $A$ must be semisimple. ...
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176 views

Intuition behind restriction and extension of scalars of group rings

Suppose we have a finite group $G$ with subgroup $H\leq G$. If $R$ is a commutative ring then we have the group rings $B=R[G]$ and $A=R[H]$, along with the natural inclusion $i:A\hookrightarrow B$. ...
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173 views

Generalized Formal Power Series Ring

I came up with the following generalization of formal power series and wonder if anyone knows a reference where this is studied. Let $R$ be a commutative ring with unity and $M$ a commutative monoid ...
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Simple notation question: What does $(kG)^\times$ mean? $k$ is a field and $G$ is a group

I guess $kG$ is the group algebra over a finite group, i.e. the set of linear combinations of $k$ and group elements. But I do not understand the "$^\times$". Thank you.
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Help in understanding what is going on in $A[G]$

Ok, so Im given a group $G$ and a ring $A$, and define: $$A[G]=\left\{\sum_{g \in G} f(g) g : f:G \to A, \text{ such that $f$ has finite support} \right\}$$ Define the sum $(+)$: $$\sum_{g \in G} ...
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222 views

Identify all units of the group ring $\mathbb{Q}(G)$ where $G$ is an infinite, cyclic group.

This question is related to this one, in that I am asking about the same problem, but not necessarily about the same aspect of the problem. I need to identify all units of the group ring $\mathbb{Q}(...
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1answer
55 views

Semigroup algebra of an idempotent semigroup

Let us consider $S=\{1,2\}$ with the operation $xy=\max\{x,y\}$. Then $S$ is a commutative semigroup with unity. Consider its complex algebra $\mathbb C[S]={\rm span}\{e_1, e_2\}$, where $e_i e_j = e_{...
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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 + ...
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1answer
267 views

Is the group ring of a finite cyclic group a “PID”?

Let $G=\langle\sigma\rangle$ be a group wirtten multiplicatively. Assume that $|G|=n$. The group ring of $G$ (denoted by $\mathbb{Z}[G]$) is defined as $$\mathbb{Z}[G]=\{\sum_{i=0}^{n-1}a_ig^i\text{ }...
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If $P$ is a fg projective $\mathbb{Z}[G]$-module, what is $P\otimes\mathbb{R}$?

Let $G$ be a finite group, and $P$ a fg projective $\mathbb{Z}[G]$-module. By a theorem of Swan, $P\otimes\mathbb{Q}\cong\mathbb{Q}[G]^n$ for some $n$. Is the same true for $\mathbb{R}$? Clearly, $(P\...
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1answer
156 views

Prove that the group ring $\mathbb{Z}(\mathbb{Z}_{n})$ can be generated by a single element

I need to prove that the group ring $\mathbb{Z}(\mathbb{Z}_{n})$ can be generated by a single element, but I'm not really sure how to begin. I know that the group ring $\mathbb{Z}(\mathbb{Z}_{n})$ ...
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2answers
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Prove that $G$ embeds as proper subgroup into group of units of the group ring $\mathbb{Z}(G)$

I need to prove that a group $G$ embeds as a proper subgroup into the group of units of the group ring $\mathbb{Z}(G)$. The group ring $\mathbb{Z}(G)$ consists of the set of formal sums $z_{1}g_{1} + ...
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296 views

Number of elements in group ring $R(G)$ in terms of $|R|$ and $|G|$

Let $R$ be a finite ring, and $G$ be a finite group. I need to compute the number of elements of the group ring $R(G)$ in terms of $|R|$ and $|G|$ (where $|R|$ is the number of elements in the ring $R$...
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63 views

Reducible modules and representations

I've started to study groups representations theory with the book of the same title, by Larry Dornhoff (part A). In the introduction chapter, there's an observation regarding matrices that I'm not ...
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49 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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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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195 views

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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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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321 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
170 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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2answers
457 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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1answer
89 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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1answer
115 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
64 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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213 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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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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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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96 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 $...
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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:...