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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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Extending scalars from $\mathbb{Z}[G]$ to $\mathbb{Q}[G]$

Let $G$ be a finite group. Let $M$ and $N$ be finitely generated $\mathbb{Z}[G]$-modules such that $M$ is free as a $\mathbb{Z}$-module. Suppose that $\mathbb{Q}\otimes_\mathbb{Z}M$ and $\mathbb{Q}\...
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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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Group algebra is domain iff doesn't contain nonzero element whose square is 0

Let $G$ be a torsion-free group and let $K$ be a field. I have to prove that the group algebra $KG$ is an integral domain if and only if it doesn't contain a nonzero element whose square is equal to 0....
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Group ring definition

I don't understand the definition of a group ring. In "An Introduction to Group Rings" by Polcino and Sehgal (page 129), an element of the group ring $RG$ is a linear combination of elements from $G$ ...
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35 views

Wedderburn decomposition of $F(Z_2\times Z_2).$

Let $F$ be any finite field of characteristic different from $2$. I have to prove $F(Z_2\times Z_2)\cong F\oplus F\oplus F \oplus F.$ I know that $F(Z_2\times Z_2)\cong F(Z_2)\otimes_F F(Z_2)$ and $F(...
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Group rings decomposition

Let $G$ be a group with $|G| = 8$. By the Artin-Wedderburn Theorem, $\mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a ...
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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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Powers of ideals in group ring $\mathbb{Z}_nS_m$

Let the group ring $\mathbb{Z}_2S_3,$ where $S_3$ is the permutation group on $3$ elements, with presentation $S_3 = \{1, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau \} = \langle \sigma, \tau | \...
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How orbits of $G$-sets and these characters are related

I've been learning about induced representations recently and I've come across something which I'm very confused about; For any $G$-set $X$, the number of orbits is equal to $(1_G, \chi_{\mathbb{C}[...
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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 $...
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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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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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Dimension of $J(F(GL_2(3)))$, where $J$ stands for Jacobson radical.

I want to find dimension of $J(F(GL_2(3)))$, where $J$ stand for Jacobson radical, for the group algebra $F(GL_2(3))$ of general linear group of two by two matrices over the field $\mathbb{Z}_3,$ and ...
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How to do congruence-class arithmetic?

When working through this question: Write out the addition and multiplication tables for the congruence-class ring $F[x]/(p(x))$. $F=\mathbb{Z_2}$; $p(x)=x^{3}+x+1$. [Question #1 in section 5.2: ...
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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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38 views

Group ring isomorphic to matrix ring

I want to figure out whether $R$ is isomorphic to $S$, where $R = \mathbb{R}[G]$, where $G = \mathbb{Z}/2 \times \mathbb{Z}/2$, and $S = M_4(\mathbb{R})$. It seems that they might not be isomorphic, ...
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Showing that three rings are isomorphic

Let $C_3 = \langle a | a^3 = e \rangle$ and let $R=(\mathbb{Z}/2)[C_3]$ be the group ring of $C_3$ with $\mathbb{Z}/2$ coefficients. Let $S = (\mathbb{Z}/2)[y]/(y^3-[1])$ and let $T = \mathbb{Z}[x]/(...
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$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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1answer
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Description of the group $1+J(FG),$ where $J(FG)$ is jacobson radical of the group ring $GF.$

My group is $G=(\mathbb{Z}_3\times\mathbb{Z}_3)\rtimes\mathbb{Z}_3$ which is non abelian group of order $27.$ Now my problem is whether the group $1+J(FG)$ is abelian or non-abelian and what is its ...
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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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The group ring of a ring.

Let $R$ be a ring. Since $R$ is also a group then we can talk about the group ring $R[R]$. I want to understand this group ring $R[R]$. An element $x\in R[R]$ is written as a finite formal sum $$x=...
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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 requested: injective modules and subgroups

Let $G$ be a group, and let $H$ be a subgroup of $G$. Suppose that $E$ is an injective module over $\mathbb{Z}G$. Then I think that, as a consequence of Baer's criterion, $E$ is also an injective ...
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25 views

Reconstructing a group from group ring and augmentation map

I've been studying Kähler differentials, and in particular there is the universal derivation, which goes hand in hand with a module. Say that we have an $A$-algebra called $B$, then we get a universal ...
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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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1answer
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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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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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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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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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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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1answer
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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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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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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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1answer
63 views

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
61 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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123 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 ...
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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)...