Recently Active 'ra.rings-and-algebras+gr.group-theory' Questions - Page 4

2 votes

0 answers

96 views

Could we assume without loss of generality that all coefficients are positive?

Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...

MSMalekan

2,118

asked May 28, 2019 at 12:27

3 votes

1 answer

158 views

Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]

I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...

spin

2,821

answered May 16, 2019 at 20:16

6 votes

0 answers

255 views

Homotopy quotient of groups

Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups. We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$. What ...

Lukas Woike

1,382

modified May 16, 2019 at 11:36

2 votes

1 answer

310 views

Flatness of submodules of free modules

Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group. If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...

Phil Tosteson

3,938

answered May 3, 2019 at 3:44

13 votes

2 answers

921 views

The set of orders of elements in a group

Let $A$ be a subset of natural numbers. Consider the following problem: Is there a group $G$ such that $\lbrace O(x) \; | \; x \in G \rbrace = A\cup\lbrace 1\rbrace$ ? (where $O(x)$ is the order of $...

Stefan Kohl♦

19.6k

modified Apr 7, 2019 at 15:38

5 votes

1 answer

605 views

Unipotent completion of free group

Whilst I am reading articles on unipotent completion to understand its basic construction, I found something confusing. Let $F$ be a free group of rank 2 whose generating letters are $x$ and $y$ and ...

Adrien

8,524

answered Apr 3, 2019 at 8:28

2 votes

1 answer

229 views

Has the "semidirect monoid of a semiring" been considered anywhere?

Given a semiring $S$, we get a monoid $M(S)$ as follows: The underlying set of $S$ is $S^2$ The identity element is $(0,1)$ The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...

Zhi-Wei Sun

15.6k

modified Mar 15, 2019 at 22:24

5 votes

0 answers

187 views

Any f.p. faithful simple module over a primitive group ring?

Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$. There ...

YCor

63.9k

modified Mar 8, 2019 at 13:27

38 votes

0 answers

1k views

Groups whose complex irreducible representations are finite dimensional

By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting. It is easy ...

Martin Sleziak

4,713

modified Dec 31, 2018 at 9:19

1 vote

0 answers

71 views

Non-zero homomorphism from a module to its ground ring

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\...

Alireza Abdollahi

3,738

asked Dec 13, 2018 at 14:57

1 vote

0 answers

31 views

Defect of subnormality in unit groups of modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. $G$ is normal in $1+rad(KG)$ if and ...

Sven Wirsing

581

asked Nov 25, 2018 at 17:04

20 votes

4 answers

3k views

Relationship between the cohomology of a group and the cohomology of its associated Lie algebra

Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...

Bonbon

806

modified Nov 24, 2018 at 16:57

1 vote

1 answer

113 views

Does this element belong to all powers of the augmentation ideal of the group algebra.

Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $...

MSMalekan

2,118

answered Nov 21, 2018 at 6:35

6 votes

1 answer

360 views

Zero divisors in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...

Alex M.

5,407

modified Oct 24, 2018 at 13:00

1 vote

0 answers

185 views

Unitary element of the group algebra

Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...

MSMalekan

2,118

asked Oct 21, 2018 at 10:57

3 votes

0 answers

109 views

Does this element belong to $\mathbb CG$?

Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...

YCor

63.9k

modified Oct 21, 2018 at 10:38

5 votes

0 answers

296 views

Are these element in a group algebra of a torsion-free group zero divisors?

Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements‌ can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)? $$1+x+y,\quad 4+x+x^{-1}+y+...

YCor

63.9k

modified Oct 10, 2018 at 16:48

7 votes

1 answer

425 views

Extension-field subgroups of $\operatorname{GL}(n, K)$

$\newcommand{\op}[1]{\operatorname{#1}}\def\GL{\op{GL}}$I am interested in the subgroups of $\GL(n,K)$ of the form $\GL(n/s, E)$ for $E$ some extension field of $K$ of degree $s$. These arise in the ...

Sean Eberhard

9,627

answered Aug 7, 2018 at 14:10

4 votes

0 answers

218 views

Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$. In fact, it's not important that I have exactly one representative from ...

Frieder Ladisch

6,919

modified Jul 13, 2018 at 11:29

27 votes

3 answers

3k views

Is there a 'nice' interpretation of virtual representations?

Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...

John Baez

22.3k

modified Jul 12, 2018 at 21:38

14 votes

2 answers

863 views

Example of a ring with non-finitely generated unit group?

The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...

YCor

63.9k

answered Jul 12, 2018 at 19:16

5 votes

1 answer

271 views

Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings?

Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a ...

Uri Bader

11.6k

modified May 31, 2018 at 11:10

1 vote

0 answers

397 views

A functor on the category of commutative rings, algebras or Banach algebras

Edit: According to the comments of abx and Yemon Choi I revise the question as follows: Let $G$ be a group and $\mathcal{A_G}$ be the category of $G$-module commutative algebras, that is the ...

Ali Taghavi

356

modified Apr 25, 2018 at 19:19

10 votes

1 answer

274 views

A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids

In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following. ...

CommunityBot

1

modified Apr 12, 2018 at 2:35

7 votes

2 answers

485 views

Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring

Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" ...

CommunityBot

1

modified Apr 5, 2018 at 19:06

9 votes

1 answer

521 views

Which group algebras in analysis are "true group algebras"?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that $$ \pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...

Sergei Akbarov

7,426

modified Mar 23, 2018 at 9:12

2 votes

0 answers

127 views

Multiplicative subgroups of $GL(V)$ which are almost additively closed

Edit: According to comments of YCor and Vincent, I revise the question.I appreciate their comments: Let $G$ be a group. We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...

CommunityBot

1

modified Feb 9, 2018 at 9:12

10 votes

1 answer

1k views

Equivalent descriptions of Coherent Groups

Attending a series of lectures, I have recently been exposed to the notion of Coherent groups, defined as following: Def: A group $G$ is called Coherent if every finitely generated subgroup $H$ of $G$...

CommunityBot

1

modified Jan 19, 2018 at 17:54

2 votes

1 answer

255 views

Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"?

The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to Google Scholar, it's been cited 233 times, and I keep seeing ...

j.c.

13.6k

answered Jan 5, 2018 at 20:02

5 votes

1 answer

548 views

Question about denoting/designating of algebraic structures

I saw this image on Wikipedia (Template:Group-like structures, current revision): Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, ...

Jacques Carette

11.8k

answered Dec 18, 2017 at 13:36

-1 votes

1 answer

218 views

A Rng of rotations?

It's very standard to view rotations about the origin in $\mathbb{R}^2$ as a group $\mathbb{SO}(2)$, with the zero rotation as an identity and composition of rotations as addition. This can also be ...

Julian Rosen

9,061

answered Dec 3, 2017 at 23:12

1 vote

0 answers

155 views

Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem

Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...

Sven Wirsing

581

modified Nov 20, 2017 at 22:14

3 votes

0 answers

65 views

Intersections of generating sets of subalgebras

Let $A$ be a finitely generated, finitely presented, Noetherian, unital algebra over the complex numbers, which has no zero divisors. We do not assume that $A$ is commutative however. Moreover, let $...

Hans gluckmann

53

modified Oct 22, 2017 at 12:54

8 votes

0 answers

685 views

What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?

Groups Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$. Question 1. What classifies involutive automorphisms on a given (non-...

Manuel Bärenz

5,565

modified Sep 13, 2017 at 8:47

8 votes

2 answers

255 views

Finding a compatible multiplication for a given group

If you are given an abelian group $\ (G, +)$, is there some algorithm to find all possible semigroups $\ (G, ×)$, such that $\ (G, +, ×)$ is a ring? If not, can you at least decide, if the ring must ...

Keith Kearnes

14.6k

modified Sep 6, 2017 at 4:37

3 votes

1 answer

137 views

Subalgebras with finite codimension

In group theory it is well-known that every subgroup of finite index contains a normal subgroup of finite index. It is not true in general that for Lie algebras every subalgebra of finite codimenslon ...

Ali Taghavi

356

modified Aug 27, 2017 at 1:24

36 votes

17 answers

6k views

Canonical examples of algebraic structures

Please list some examples of common examples of algebraic structures. I was thinking answers of the following form. "When I read about a [insert structure here], I immediately think of [example]." ...

François G. Dorais

44.4k

modified Aug 9, 2017 at 1:35

4 votes

1 answer

434 views

Regarding a new algebraic structure

By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this ) $$ x*(y\cdot z)=x*y*z\;\; ; \;...

François G. Dorais

44.4k

modified Aug 9, 2017 at 0:23

4 votes

1 answer

154 views

Right reversibility of submonoids of nilpotent groups

Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$). A (cancellative) monoid/semigroup $S$ is right reversible if for ...

Benjamin Steinberg

38.6k

modified Jul 29, 2017 at 18:24

6 votes

2 answers

483 views

In the group ring $\mathbb{Z}_p [G]$, what elements satisfy $(\sum a_g g)^p = \sum a_g g^p$?

Here $\mathbb{Z}_p$ is the ring of integers in $\mathbb{Q}_p$. Preferably I would want to know this for a general group $G$, but I have been concentrating on the case $G = (\mathbb{Z} / p^n \mathbb{Z}...

Geoff Robinson

44.4k

modified Jun 7, 2017 at 12:51

9 votes

1 answer

893 views

Kaplansky conjecture (consequences)

The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements. Questions Do we assume that $F$ has any ...

Zahra Taheri

632

modified May 24, 2017 at 21:18

27 votes

5 answers

3k views

Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?

Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...

David Handelman

4,679

answered May 14, 2017 at 15:18

16 votes

3 answers

3k views

Are there other semidirect product/crossed products in other areas

Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only ...

CommunityBot

1

modified Apr 4, 2017 at 22:12

0 votes

1 answer

75 views

How do we prove that the following implication in semiring? [closed]

Let $G$ be a group. Clearly the power set $(\mathcal{P}(G),\cup,. )$ is the semiring, where $\cup$ means ordinary union and '.' is defined as $$AB= \left\lbrace ab \in G \mid a\in A\mbox{ and } b\in B ...

Jochen Wengenroth

16.4k

modified Mar 22, 2017 at 7:00

2 votes

3 answers

545 views

Irreducible representations of $\text{SL}(2, \mathbb{F}_q)$ which don't exist in decomposition?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0$. ...

Peter McNamara

8,866

modified Mar 22, 2017 at 0:05

2 votes

1 answer

151 views

Right socle of a group ring

Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z_{(p)}C_n$, where $\mathbb Z_{(p)}$ is the localization of integers at the ...

Jeremy Rickard

35.2k

answered Mar 13, 2017 at 8:53

6 votes

2 answers

210 views

groupring morphisms and bialgebra

Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...

Ender Wiggins

718

answered Mar 8, 2017 at 14:20

24 votes

3 answers

1k views

Is $\widehat{\mathbb{Z}}[[t]]\cong\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]$?

Let $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]] := \varprojlim_{n,m}(\mathbb{Z}/n)[x]/(x^m-1)$ be the complete group algebra of the profinite free group of rank 1. In Corollary 5.9.2 of Ribes-...

CommunityBot

1

modified Feb 28, 2017 at 22:43

13 votes

2 answers

515 views

Free groups and free restricted Lie algebras

If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows $$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...

darij grinberg

33.8k

modified Feb 4, 2017 at 5:44

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