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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How "exotic" are non-local directly irreducible modular group algebras?
Let $G$ be a finite group and $R = \Bbb F_{p^k} G$ a modular group algebra ($p$ divides $|G|$). I would like to know when $R$ is a directly indecomposable ring. It is quite well-known that $R$ is ...
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Has anyone studied differential equations on $\mathbb{Z}F_n$ defined using Fox derivatives?
I am looking for a reference, if it exists, to the study of differential equations defined using Fox derivatives over the group ring, say, of a free group. Is this a topic which has been studied ...
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Module structure on $\mathbb{Z}$ over the Groupring $\mathbb{Z}[G]$
I would like to know how exactly the module structure on $\mathbb{Z}$ as a module over the group ring $\mathbb{Z}[G]$ works.
Obviously, we take just the usual addition on $\mathbb{Z}$, but how does ...
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Is there a way to determine when an abelian group $(M,+)$ admits a $R[G]$-module structure, where $R[G]$ is a group-ring.
As the title says, are there any relevant theorems, which determines when, given an abelian additive group $(M,+)$, we can determine whether $(M,+)$ admits a $R[G]$-module structure, for a unital ring ...
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Group ring zero divisor example.
Here is the question I am trying to understand its solution:
Give an example of a zero divisor in the group ring $\mathbb Z(C_3)$ where $C_3$ is the cyclic group with 3 elements.
My professor gave us ...
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Every $K[G]$-module is torsionless?
Let $G$ be a finite group and $K$ a field. Consider the group ring $R:=K[G]$. Let $M$ be a (left) $R$-module. Is it true that then there exists a set $S$ and an injective $R$-module homomorphism
$M\...
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What is the $\mathbb{F}_p$ dimension of $\mathbb{F}_p[G]$?
I heard recently that the $p$-rank of a finite abelian group (the number of cyclic components of size $p^n$) is given by $\dim_{\mathbb{F}_p}(\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G])$, which is ...
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Some calculation in $F_5[D_{30}]$.
I am currently reading a research paper and have encountered a point that I am struggling to understand. In the paper, it is proven that $J(F_5[D_{30}])^5 = (0)$, indicating that the Jacobson radical ...
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What is the monoid ring $K[(\Bbb{N}, \text{lcm})]$ isomorphic to?
$(d\mid \cdot)(c\mid \cdot) = (\text{lcm}(d,c) \mid \cdot)$ where $(n\mid x) \in \{0,1\}$ is whether (1) or not (0) $n$ divides $x \in \Bbb{Z}$.
Thus, we can linearly extend all formall $K$-linear ...
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For which groups $G$ there is an integer $n$ with this property?
Let $G$ be a finitely presented group and $M$ be a finitely generated $ZG$-module. By a retract of $M$, I mean a submodule $N$ of $M$ for which there is a homomorphisms $f:M\to N$ such that $fi=id_N$, ...
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Preimage of a maximal ideal is maximal in free ring extensions
Suppose that $A, B$ are commutative rings with identity, and we
have an injective ring homomorphism $A \hookrightarrow B$.
I would like to know wether or not the following proposition is true:
If $B$ ...
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Group ring $\mathbb{Z}G$ with special property on $G$ [closed]
Let $H$ be a subgroup of $G$. Then a homomorphism $r:G\to H$ is said to be a retraction if the inclusion homomorphism $i:H\hookrightarrow G$ is a right inverse of $r$, i.e. $r(x)=x$ for all ...
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On the Wedderburn component of a group algebra.
Let $G$ be a finite group and $\mathbb{F}_q$ be a finite field, where $q$ is a prime power.
Let $\mathbb{F}_qG$ be a finite semisimple group algebra.
Then Wedderburn decomposition theorem implies that ...
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Group Ring $F(G\rtimes H)$ .
Is there any relation between group rings $F(G\rtimes H)$ and $F(G)\rtimes F(H)$, where F is a finite field and $\rtimes$ is semi direct product of finite groups $G$ and $H$?
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$End_k(M ⊗_k K) \simeq K$
Let $G$ be a finite group, $k$ be an algebraically closed field such that $char(k)\nmid |G|$ and K/k be a field extension.
If M is any simple $k[G]$-module show $End_k(M ⊗_k K) \simeq K$
What I've ...
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Group Ring epimorphism.
I am reading a research paper in which group $G=D_{30}=\{x,y\mid x^{15}=y^2=1, yxy=x^{-1}\}$ and $K$ be the normal subgroup of $G$ generated by $x^5$. Then $G/K\cong H\cong \langle x^3,y\rangle.$ ...
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Group ring over a cyclic group
We have the group ring $R= \mathbb{Z} C_5$ over the cyclic group of order 5, generated by the element $g$.
Find an $a,b \in R$ such that $ab=0$ with $a,b \neq 0$
Since 0 is not in the cyclic group, ...
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Onto map between the group algebras.
Let $\mathbb{F}_p$ be a finite field of order $p$ ($p$ prime) and $G=SL(2,7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F_7}.$ Let $\mathbb{F}_pG$ be the group ring (or ...
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Hasse invariants of Wedderburn components of group rings attached to a group and a normal subgroup
Let $G$ be a finite group and $H$ a normal subgroup. Let $F$ be a local field of characteristic zero. The group algebras $F[H]\subset F[G]$ are semisimple by Maschke’s theorem. Consider their ...
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Representations of the centre of the Group ring
Given a finite group $G$, let $\rho:G \to GL(n,\mathbb{C})$ be an irreducible representation of $G$.
Now consider the group ring $\mathbb{C}G$. There is a very natural way of extending $\rho$ to $\rho^...
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Image of augmentation ideal of a group ring under ring homomorphism.
Let $G,H$ be two finite $p$-groups and $f:\mathbb{F}_p[G]\to \mathbb{F}_p[H]$ a ring homomorphism of mod $p$ group rings.
Is it always true that $f(I_G)\subset I_H$, where $I_G,I_H$ are the respective ...
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Description of projective $\mathrm{SL}_{2}(\mathbb{Z})$-modules
I am working my way through Ken Browns book on the cohomology of groups, and in particular chapters 8 and 9 on finiteness conditions, and Euler characteristics.
Most of the concepts in chapter 9 (such ...
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Recovering original definition of group cohomology from Ext definition
I've recently been studying group cohomology, the original definition I learned was that of Ext, where $H^n\left(G, M\right)= \text{Ext}_{\mathbb{Z}G}^i\left(\mathbb{Z}, M\right)$. I then read a ...
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Structural description of the center $Z(R[G])$ of a group ring
Let $G$ be a finite group and $R$ be a ring. There is a well-known description of the center of the group ring: $Z(R[G])$ is a free $Z(R)$-module, the basis consists of all $\sum_{g \in K} g$, where $...
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Jacobson radical of group ring is $\{\sum a_gg =0:\sum a_g =0\}$
Let $G$ be a finite group and $F(G)$ the group ring over the finite field $F$.
Prove $Jac(F(G))=\{\sum a_gg =0:\sum a_g =0\}.$
I know that $Jac(F)=0$. Is there a possibility to show $Jac(F(G))=Jac(F)...
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Is the direct sum preserved under isomorphism
I have edited this based on discussions in the comments:
This really is more of a sanity check and I couldn't seem to find an answer to anything similar anywhere, so apologies if this has already been ...
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Decompositions of a group algebra
I am a math student just starting to know group algebra. Let $FG$ be a group algebra in which $F$ is a field and $G$ is a group. I am particularly interested in the results of the decomposition of $FG$...
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About the definition of group algebra
In several texts it is standar to define the group algebra $\mathbb{C}[G]$ of a group $G$ as
$$
\mathbb{C}[G]:=\left\{ \sum_{x\in G}c_{x}x \colon c_{x}\in \mathbb{C} \right\}.
$$
Now, these ...
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Mistake in proof: Symmetric group acting on group ring
Let $G$ be finite group of order $n$ and $K$ be a field. The symmetric group on $G$, denoted by $S_G$ acts linearly on the group ring $K[G]$ by
$$\sigma\left(\sum_{g\in G}c_gg\right):=\sum_{g \in G}...
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About the intersection of two commutator subgroups
I'm reading an example in Lennox & Stonehewer's book "Subnormal Subgroup of Group".
There (p.144/145) they construct the following example. Let $L$ be an infinite elemetary abelian $2$-...
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Isomorphism between $\mathbb{Z}[C_n]$ and $\mathbb{Z}[X]/(X^n-1)$ [duplicate]
I'm trying to prove that $\mathbb{Z}[C_n] \cong \mathbb{Z}[X]/(X^n-1)$ where $C_n$ is the cyclic group of order $n$. Let $t$ be a generator of $C_n$. There is a unique ring homomorphism
$$\Phi:\mathbb{...
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Representation of $\mathfrak{S}_p$ over $\mathbb{F}_p$
I'm learning modular representation theory from the 3rd part of Serre's book. Through the process called "reduction mod. $\mathfrak{m}$", we obtain representations in positive characteristic ...
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$\mathbb{Z}$ as $\mathbb{Z}G$ - module.
Let G group. Recently i read that $\mathbb{Z}$ can be considered as $\mathbb{Z}G$- module with the trivial action, such that :
$$\forall x\in \mathbb{Z}G,\ z\in \mathbb{Z}:\quad x\cdot z:=z$$
But if ...
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Is (3) a prime ideal of $\mathbb{Z}$ $(\sqrt{-17})$
First, I had to prove that $\mathbb{Z}$ $(\sqrt{-17})$ is not a UFD which was done by showing that in general $\mathbb{Z}$ $(\sqrt{-n})$ for $n \geq 3$ is not a UFD. Subsequently, I was asked if (3) ...
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Lie algebra structure on $\mathbb{C}[G]$
Let $G$ be a finite group. The group algebra $\mathbb{C}[G]$ has a Lie algebra structure inherited by its associative algebra structure, which is given by
$$[g,h]=gh-hg.$$
It is not too hard to check ...
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Group Ring/Algebra and Functions of Positive Type
Given a group $G$, we can form its complex group ring/algebra $\Bbb{C}[G]$, which either be described as the set of all finite formal sums of group elements with complex coefficients or as all the ...
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Homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$
Show that there are at most $4$ ring homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$.
Here is what I did :
We know that ring homomorphisms send inversible elements to ...
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Doubt about invertibility in Group Algebra over a cyclic group
Let $C=\langle c \rangle$ denote the cyclic group of order $2$ written multiplicatively, i.e. $C$ is a copy of $\{\pm 1\}$ with the usual product, and let $k$ be a field of characteristic different ...
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Generators for $\mathbb{C}[G](1-g) \cap \mathbb{C}[G](1-h)$ given by relations in $\langle g,h \rangle$?
Let $G$ be a group and $g,h \in G$ two elements. I want to describe a set of generators for the left ideal $\mathbb{C}[G](1-g) \cap \mathbb{C}[G](1-h)$.
Suppose we have a relation $$g^{n_1}h^{m_1}g^{...
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Isomorphism between $\overline{\mathbb{C}[G \times G]}$ and $\overline{\mathbb{C}[G]} \otimes_{min} \overline{\mathbb{C}[G]}$
I'm currently reading through a paper and it has a sort of throwaway line that says that we have the following isomorphism at our disposal:
$\otimes_{k=1}^{n} \overline{\mathbb{C}[G]} \simeq \overline{...
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Why is an abelian group an algebra over $GF(2)$?
In this MO answer, Geoff Robinson writes:(Punctuation error corrected)
Consider a group homomorphism $\phi:H\to A$, where $A$ is an Abelian group. Let $R$ be the group ring $GF(2)[A]$. Consider $\phi$...
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Calculations in a group ring with computer algebra system
I am currently trying to read the paper Constructions of difference sets by Applebaum et al. Unfortunately, there are some very elaborate calculations. For example, in example 1.13 in the group ring $\...
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Polynomials with exponents in numerical semigroup
When thinking about monoid algebras (semigroup rings), I came across the following:
Let $S$ be a numerical semigroup, i.e., an additively closed subset of $\mathbb{N}_0$ containing $0$ such that $\...
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3
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Express a function $g$ in the vector space $\mathbb{Z}/3$
I have a question I can't seem to solve. It is stated as follows:
Recall that $\mathbb{Z}$ denotes the ring of integers, and $\mathbb{Q}$ is the rationals. We let $F[x]$ denote the polynomial ring in ...
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No onto map between the algebras.
I want to prove that there does not exist a surjective group algebra homomorphism from $FS_5$ (the group algebra (or group ring) of the symmetric group, $S_5$, over the field $F$) to $M_2(F)$, where $...
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Could the discovery of a counterexample to the Unit Conjecture change mathematicians’ understanding of spinors in general?
Recently it has been reported that Giles Gardham has found a counterexample to the Unit Conjecture for group rings, as given in https://arxiv.org/abs/2102.11818 (and also in a popular article https://...
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Tensor products of group algebras
We know that if $G$ and $H$ are finite groups and $F_p$ is a field of characteristics $p$, then $$F_pG\otimes_{\mathbb{F}_p} F_pH\cong F_p(G\times H).$$ Here $\otimes_{\mathbb{F}_p} $ denotes the ...
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Module Over the Group Algebra of a Ring
I'm aware that if $K$ is a field, $V$ a $K$-vector space, and $G$ is a group, then a $K$-representation of $G$ in $V$ is the same thing as a $KG$-module, where $KG$ is the group algebra of $G$ over $K$...
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1
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Why that element belongs to FC subgroup?
Let $\Delta$ denote Finite Conjugate subgroup of group G. K - a field
Then
The photo above is from Passman book "Infinite group rings".
I dont understand why elements $u_{i}$ belong to $\...
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Isomorphism Between Cyclic group algebra and Quotient Polynomial Ring
Let $F$ be a field. I'd like to show that:
$a)$ If $V$ is a cyclic group of order $n$, then $$F[V] \cong F[x]/(x^n - 1).$$
$b)$ If $\operatorname{char}(F)\neq 2$, $V=\{1, f, g, fg\}$ is the Klein ...
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