All Questions
Tagged with ra.rings-and-algebras gr.group-theory
309 questions
18
votes
1
answer
783
views
Are there any "simple" monoids with intermediate growth?
The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...
- 1,947
9
votes
1
answer
580
views
Units of group algebra of dihedral group
Question:
Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite ...
- 63.9k
2
votes
0
answers
81
views
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 $...
- 12.9k
0
votes
1
answer
426
views
Generators of $SL(n,\mathbb F_2)$? [closed]
Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...
- 37.4k
1
vote
1
answer
344
views
Bound for order of a group depending on number of elements of maximal order
This question has been partly answered in MSE, see here.
In a paper On the Number of Elements of maximal order in a Group, it is proven that an arbitrary group $G$ with a finite number of elements of ...
- 181
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
3
votes
1
answer
355
views
The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
- 148k
2
votes
0
answers
111
views
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
5
votes
0
answers
424
views
A lemma concerning conjugations and normal subgroups (related to a theorem of Frobenius)
In the paper "On A Theorem of Frobenius" written in 1969, Prof. Richard Brauer, for the first time, presented a character-free proof to the Frobenius theorem (i.e. counting the number of ...
- 63.9k
1
vote
1
answer
275
views
The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that
$$
\U(2^{N-1})\supset \Spin(2N)
$$
when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
- 108k
21
votes
10
answers
4k
views
Why do we associate a graph to a ring? [closed]
I don't know if it is suitable for MathOverflow, if not please direct it to suitable sites.
I don't understand the following:
I find that there are many ways a graph is associated with an algebraic ...
- 25.8k
12
votes
1
answer
2k
views
Relationship between the Witt algebra and vector fields on the circle
I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.
The ...
CommunityBot
- 1
1
vote
3
answers
1k
views
What are the main open problems in Group Theory and Galois Theory? [closed]
I’m really interested in doing a PhD and the subjects I enjoyed the most were Group Theory and Galois Theory.
What are some open problems in these areas that would be suitable for a PhD? Is Galois ...
- 12.9k
28
votes
1
answer
2k
views
Integer matrices which are not a power
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
- 2,821
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] ...
- 63.9k
8
votes
0
answers
251
views
When does a semisimple $\mathbb{C}$-algebra come from a group?
Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras:
$$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$
...
- 297
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
28
votes
1
answer
2k
views
Does GL_n(Z) have a noetherian group ring?
Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature?
Motivation: a recent trend has been to study "...
- 63.9k
5
votes
1
answer
132
views
Integral monoid rings and Ore conditions
Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.
I have two, ...
- 18.7k
2
votes
1
answer
160
views
MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
- 37.4k
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?
- 63.9k
6
votes
2
answers
235
views
Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?
Let $k$ be a finite field with char$(k)=p>0$. Let $G$ be a finite group.
Consider the group algebra $kG$.
I would like to solve a given system of equations in $kG$.
Question:
Is there a computer ...
- 5,898
18
votes
4
answers
2k
views
For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ ...
- 63.9k
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
12
votes
1
answer
584
views
Unit group of octonions over finite fields
One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: https://www.gap-system.org/Manuals/doc/ref/chap62.html .
When $K$ is a ...
- 85.6k
3
votes
0
answers
180
views
Units in group rings
Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
- 63.9k
3
votes
2
answers
279
views
Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
Let $G$ be a finite group. Let $p$ be a prime.
Let $O_p(G)$ be the $p$-core of $G$.
Are there any theorems known saying something like
$O_p(G)$ is trivial, if and only if ... and
$O_p(G)$ is non-...
- 5,407
6
votes
0
answers
294
views
Independence results on pure algebra
I think that the most celebrated result in this direction is Shelah's famous work on Whitehead's Problem:
Is every abelian group $A$ such that $Ext^1(A, \mathbb{Z})=0$ free?
This is known to be ...
- 3,318
3
votes
1
answer
161
views
Behavior of invariants under reduction mod p
Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a ...
- 3,504
3
votes
1
answer
95
views
Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks
A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
- 44.4k
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 ...
- 63.9k
3
votes
1
answer
142
views
How to identify the two copies of $D_{24}$ in the homomorphisms of the 2 musical actions? [closed]
Let $S$ be the set of minor and major triads. Two sets of actions are defined on the set:
1) Musical transposition and inversion;
2) P, L, R actions
$$P(C-major) = c-minor,$$
$$L(C-major) = e-minor,$...
- 2,175
15
votes
4
answers
869
views
What is known about ordinary character values at involutions?
Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...
- 44.4k
1
vote
1
answer
95
views
Local submodules of finite rings
Let $A$ be a finite local ring $R$, such as a group algebra of $p$-groups over a finite field of characteristic $p$.
Question 1: Is it possible, using GAP, to obtain the poset of all submodules of $R$...
CommunityBot
- 1
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
3
votes
1
answer
489
views
augmentation ideal is always finitely generated?
$G$ is a finitely presented group (but not a finite group), and $\mathbb{Z}G$ is the corresponding group ring.
$I$ is the kernel of the augmentation morphism $\mathbb{Z}G\rightarrow \mathbb{Z}$.
Is $...
- 38.6k
24
votes
2
answers
3k
views
Does any textbook take this approach to the isomorphism theorems?
Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______________, but groups will get the points ...
- 4,713
9
votes
3
answers
1k
views
Does "finitely presented" mean "always finitely presented", considered in general
I'm wondering about the question
"If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?"
I know this is true for groups and ...
- 63.9k
15
votes
1
answer
566
views
Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$
Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)?
Notice that $\Bbb Z$ is not cancellable, so
$A \oplus \Bbb Z \...
- 63.9k
94
votes
2
answers
7k
views
$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
- 63.9k
6
votes
1
answer
299
views
Can a compact object be a nontrivial self-retract?
Let $\mathcal C$ be a locally finitely-presentable category, and let $X$ be a finitely-presentable object of $\mathcal C$.
Question: Can there exist a nontrivial idempotent on $X$ whose fixed points ...
- 63.1k
1
vote
1
answer
195
views
Strongly graded algebras with no zero divisors
Let $A = \bigoplus_{i \in \mathbb{Z}} A_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that
$$
m: A_i \otimes_{A_0} A_j \to A_{i+j}
$$
is an ...
- 7,746
4
votes
1
answer
199
views
Groups for which all projections of $C^*_{\text{red}}G$ belong to $\mathbb{C}G$
Revision: According to comment of Wojowu we give a complete revise for this post.
A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{...
- 1,540
0
votes
1
answer
185
views
A subset (or subgroup) associated to a group
Edit: According to comment conversations we revise the question.
Let $G$ be a group. We consider the following subset of $G$:
$$\{g\in G \mid e^{\lambda_g} \in \mathbb{C}\lambda (G)\},$$
where $\...
- 63.9k
3
votes
1
answer
393
views
Kaplansky Idempotent conjecture and Extension theory
We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...
- 356
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 63.9k
0
votes
0
answers
124
views
What do we know about this ideal of the group algebra?
Let $G$ be a torsion-free amenable group. Consider, $\mathbf M$, the collection of all multiplicative functionals on $\mathbb CG$, the complex group algebra of $G$. So, $\ker\phi$ is an ideal of $\...
- 25.8k
10
votes
4
answers
1k
views
Groups and rings which are not sets
An algebraic structure such as a group, ring, field, etc. is usually defined to be a set with some operations satisfying certain properties. I am curious what, if anything, goes wrong when the ...
- 15.4k
1
vote
0
answers
156
views
Does $AA^{-1}$ have the unique product property?
Let $A$ be a finite subset of a torsion free group $G$ with $|A|\geq3$. Does the set $AA^{-1}$ have the "unique product" property (i.e. there exist an element $c\in AA^{-1}$ that is uniquely written ...
- 2,118
4
votes
1
answer
415
views
Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$
Edit: According to the comment of Todd Trimble, I revise the question.
What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
- 356