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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$G$ finite group and $\mathbb{C}[G]$ its group ring, show: $K_i = \sum_{a_k\in cl_G(a_i)} a_k $, then $Z(\mathbb{C}[G])=span_\mathbb{C}(K_1,…,K_n)$ [duplicate]

Given a finite group $G = \{a_1 ,a_2,...,a_n\}$ and $\mathbb{C}[G]$ the respective group ring, $\mathbb{C}[G] = \{\sum z_ia_i : z_i \in\mathbb{C} , a_i\in G\}$. Defining $K_i := \sum_{a_k\in cl_G(a_i)...
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Finding idempotents in group algebra over $A_n$

Let $G=A_4$ be the alternating group on 4 letters, and let $R = \mathbb{C}[G]$. Then $$\mathbb{C}[G] = U\oplus U' \oplus U'' \oplus V^{\oplus 3},$$ where $U,U',U''$ are the three 1-dimensional ...
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Analogue for Group Rings for multiplicative sets

Suppose we have a ring $R$ and a multiplicative set $S$. Can we define $S$ rings analogous to Group Rings? I think we'll have some problems if inverses do not exist in $S$, since when multiplying ...
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When is a Hopf Algebra isomorphic to a group ring k[G]?

Let $H$ be a Hopf algebra over a field $k$. What are some nice conditions for when $H$ is isomorphic to $k[G]$ for a finite group $G$? The co-multiplication structure on the group algebra $k[G]$ is ...
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Question on Group ring $\mathbb{Q}G$ where $G$ is a finite group

Let $G=\{e=g_1,g_2,.....g_n\}$ be a finite group of order n and let $\mathbb{Q}G$ be the group ring. Let $\sigma=\sum_1^ng_i$. Prove that $\sigma^2=n\sigma$ and deduce that $\mathbb{Q}G$ has a ...
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Reference for Kaplansky's proof that in $\mathbb{C}[G]$, $ab=1$ implies $ba=1$

Here on the Wikipedia page for Group Rings, talking about group rings over infinite groups, The case [of a group ring $R[G]$ where $G$ is an infinite group] where $R$ is the field of complex ...
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The elements of the group ring $R = \mathbb{Z}_4\mathbb{C}_2$

Let $\mathbb{C}_2$ denote the group of order $2$ and let $\mathbb{Z}_4$ denote the ring of integers modulo $4$. The group ring $R = \mathbb{Z}_4\mathbb{C}_2$ is commutative. My problem is how to ...
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Isomorphism of group rings over finite cyclic groups

Let $R$ be a commutative ring with unity. For any group $G$, let the group ring be denoted by $R[G]$. If $R[\mathbb Z/(n) ] \cong R [\mathbb Z / (m)]$ as rings , then is it true that $m=n$ ? If that ...
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Can the averaging of a linear arrow of modules over a group algebra be described functorially?

Wikipedia outlines a neat proof of Maschke's theorem from the module theoretic perspective. The fundamental idea seems to be "averaging". Proposition. Write $U:\Bbbk G$-$\mathsf{Mod}\to \Bbbk$-$\...
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Classification of homomorphism as abelian group

Consider $f:\mathbb{Z}^3 \rightarrow \mathbb{Z}^4$ $$f(a, b, c) = (a+2b+8c, 2a-2b+4c, -2b+12c, 2a -4b + 4c)$$ Describe the image of this homomorphism as an abstract abelian group. Describe the ...
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Center of Group algebra finitely generated

I am going through a proof in Jean-Pierre Serre's german version of "Linear representations" and have the following theorem here. Let $\rho$ be an irreducible representation of the finite group G of ...
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Decomposition of $\mathbb{C}[G]$ / Orthogonality relations

I am currently working on Jean-Pierre Serre's "Linear representations" german translation in chapter 6 and I do not understand the last part of the proof of the following theorem. a) Let $\rho_i : G \...
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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 $(\...
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$\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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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265 views

units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic

How would I be able to describe all units of the group ring $\mathbb{Q}(G)$ where $G$ is specifically an infinite cyclic group?
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Is there any isomorphism between the quotient ring $F{_p}$ /$\left\langle {{x^n} - a}\right\rangle$ and the group algebra $F{_p}G$

We know that the quotient ring $F{_p}$ /$\left\langle {{x^n} - 1}\right\rangle$ is isomorphic to the group algebra $F{_p}C{_n},$ where $F{_p}$ is a finite field of characteristic $p$ and $C{_n}$ is a ...
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58 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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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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306 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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184 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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85 views

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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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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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 ...