All Questions
Tagged with ra.rings-and-algebras gr.group-theory
91 questions with no upvoted or accepted answers
2
votes
0
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603
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For R a ring, if $x^n=x$ for all x∈R then R is commutative
I came across the following Theorem on a paper I'm reading:
"If $R$ is a ring and $x^n=x$ for all $x\in R$ then $R$ is commutative"
However I have no idea of how to prove it. May I get some ...
- 21
2
votes
0
answers
90
views
Regular conjugacy classes and irreducible representations in the infinite, projective case
Let $k$ be an algebraically closed field and $G$ a (not necessarily finite) group. Let $\alpha\colon G\times G\to k^*$ be a multiplier, meaning that
$\alpha(s,t)\alpha(st,r)=\alpha(s,tr)\alpha(t,r)$ ...
- 1,903
2
votes
0
answers
81
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The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 10.5k
2
votes
0
answers
111
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The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $(\Spin(...
- 10.5k
2
votes
0
answers
158
views
Division rings with finitely generated group of units
Is there any classification of division rings with finitely generated group of units? Is there any non-trivial example?
- 2,233
2
votes
0
answers
238
views
Flat augmentation ideal of a group-ring
If $G$ is a group and $I$ the augmentation ideal $I=Ker(\mathbb{Z}G\rightarrow \mathbb{Z})$ suppose that:
$I$ is a flat (right) $\mathbb{Z}G$-module.
$I$ is a finitely generated (right) $\mathbb{Z}G$...
- 530
2
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0
answers
96
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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 $\beta\in\mathbb CG$ ...
- 2,118
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 $H$ ...
- 356
2
votes
0
answers
1k
views
Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
- 721
2
votes
0
answers
468
views
Semidirect product of semidirect products
For algebraic objects, say for groups $ N,K$ and algebra $A$, if we have the semidirect product of a semidirect product,
$(A \rtimes_{\gamma} N ) \rtimes_{\theta} K$, are there conditions that would ...
- 21
2
votes
0
answers
142
views
calculation in a group ring
I have some problems with the verification of the third equation in Lemma 1 in this paper.
First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ above....
- 31
2
votes
0
answers
144
views
Hall's paper on the profinite groups and Andre Weils "voisinage" notion
I am reading through a classical paper A Topology for Free Groups and Related Groups
by Marshall Hall Jr. in which profinite groups are defined for the first time.
There he defines on p. 129:
...
- 798
1
vote
0
answers
89
views
The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
1
vote
0
answers
112
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Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
- 356
1
vote
0
answers
78
views
tensor dimension/reshaping group
Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
- 11
1
vote
0
answers
74
views
The influence of the derived subgroup of the unit group of a group algebra
Let $FG$ be a group algebra in which $K$ is a field and $G$ is a group. Suppose that every element in the derived subgroup $\mathcal{U}(FG)'$ of the unit group $\mathcal{U}(FG)$ of $FG$ is a ...
- 141
1
vote
0
answers
107
views
Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
- 311
1
vote
0
answers
156
views
Does $AA^{-1}$ have the unique product property?
Let $A$ be a finite subset of a torsion free group $G$ with $|A|\geq3$. Does the set $AA^{-1}$ have the "unique product" property (i.e. there exist an element $c\in AA^{-1}$ that is uniquely written ...
- 2,118
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 $\...
- 3,738
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 ...
- 581
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 ...
- 2,118
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 ...
- 356
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 $...
- 581
1
vote
0
answers
206
views
The normalizer problem for group rings
I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\...
1
vote
0
answers
59
views
On generating Euler Square of index q, q-1 (where q is any prime power)
Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...
- 11
0
votes
0
answers
116
views
Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 10.5k
0
votes
0
answers
96
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 356
0
votes
0
answers
58
views
An action on multiplicatively antisymmetric matrix
A matrix $ Q=(q_{ij})$ is called multiplicatively antisymmetric over a field $ F $ if $ q_{ii}=1 $ and $ q_{ij}={q_{ji}}^{-1} $.Let $ \mathcal{Q} $ be the set of all $ n \times n $ multiplicatively ...
- 923
0
votes
0
answers
150
views
How to classify rings by combinatorial structures?
There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
0
votes
0
answers
250
views
Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
0
votes
0
answers
74
views
Estimate of the nilpotency class from the subgroup
Let $G$ be a nilpotent group and $H \vartriangleleft G$ a normal subgroup such that $[G:H] \le m$.
Assume $H$ has the nilpotency class $ \le n$. Can we show the nilpotency class of $G$ is bounded by a ...
- 2,535
0
votes
0
answers
141
views
Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
- 356
0
votes
0
answers
124
views
What do we know about this ideal of the group algebra?
Let $G$ be a torsion-free amenable group. Consider, $\mathbf M$, the collection of all multiplicative functionals on $\mathbb CG$, the complex group algebra of $G$. So, $\ker\phi$ is an ideal of $\...
- 2,118
0
votes
0
answers
308
views
Basic question about power series and complete group algebras
This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
- 10.7k
0
votes
0
answers
199
views
Action of semidirect products of cyclic groups
Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...
0
votes
0
answers
143
views
Soluble group algebras and centralizers
Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every $p'$-...
- 581
0
votes
0
answers
75
views
How to distinguish projective involutions?
Let $K$ be a field of characteristic not $2$ and $R$ a continuous von Neumann regular ring with centre $Z=Z(R)$ isomorphic to $K$. For an example one may assume $R$ is a matrix ring of $n\times n$-...
- 804
0
votes
0
answers
133
views
On X-s-permutable subgroups of a finite group
I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web.
This is my proof.
.
Since $H$ is $X−s−$permutable in $G$, then ...
-1
votes
1
answer
199
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Isomorphism between subgroups by preserving index
Let $C$ be a subgroup of a group $A$ such that $C\cong A/\{\pm1\}$ and $D$ be a subgroup of $B$ such that $D\cong B/\{\pm1\}$. Let $\pi:A\to B$ be a group homomorphism, and let $\pi$ induce a ...
- 309
-3
votes
0
answers
139
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103