Questions about the branch of abstract algebra that deals with groups.

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0
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0answers
47 views

Davenport constant of class group

Denote $\mathsf{C(\Delta)}$ where $\mathsf{\Delta=b^2-4ac<0}$ with $\mathsf{gcd(a,b,c)=1}$ be class group of all equivalence classes of integral quadratic forms with discriminant $\mathsf{\Delta}$. ...
3
votes
0answers
64 views

Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion: Topology on the Unitary Dual Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let ...
8
votes
6answers
781 views

What books approach group theory through transformation/permutation groups?

What are some books that discuss transformation groups (or permutation groups) before abstract groups? Some quotes to motivate the question: from V. I. Arnold, 'On Teaching Mathematics': What ...
5
votes
1answer
101 views

Zero divisors with support of size 3 in group algebras of finite groups

Are there a finite group $G$ and a field $\mathbb{F}$ such that $\gcd(3,|G|)=1$ and the group algebra $\mathbb{F}[G]$ contains a zero divisor whose support is of size $3$? Recall that the support of ...
1
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0answers
52 views

largest subgroup of $Out(\hat{F_2})$ which preserves the Nielsen invariant

Let $x,y$ be generators for the free group $F_2$. It's known that $Aut(F_2)$, and hence $Out(F_2)$ preserves the conjugacy class of the subgroup $\langle[x,y]\rangle$ generated by $[x,y]$ (This ...
1
vote
2answers
123 views

Automorphism group of the affine groups AGL(n,q), ASL(n,q)

I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...
4
votes
0answers
62 views

Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...
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0answers
128 views

A finite group with O_{p}(G)=1

Let $G$ be a finite group of order $p(p^2-1)/2$, where $p$ is prime number. If $O_{p}(G)=1$, then what is the number of Sylow $p$-subgroups G?
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1answer
167 views

Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh. The chinese remainder theorem can be stated as follows: Let $n_1, \dots, n_r \ge 2$ be positive integers ...
1
vote
1answer
214 views

Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?

$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times. I am studying some function arising from symplectic geometry which happens in my case to be ...
3
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0answers
78 views

Generalized identities of (soluble) groups

Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that $$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$ for all $x\in G$. ...
5
votes
0answers
106 views

Indecomposable representations of a wreath product

If $G$ is a finite group, we know the irreducible representations of $G ≀ S_n$ (over $\mathbb Q$) are classified by partitions of $n$ 'decorated' by an irrep of $G$. I'm wondering to what extent the ...
6
votes
5answers
2k views

Lie groups admitting flat (bi)invariant metrics.

I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric. Is there any non-abelian compact lie group admitting a flat metric that is bi or ...
9
votes
1answer
230 views

Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre. Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
26
votes
2answers
1k views

How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...
2
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2answers
118 views

Are the Baumslag-Solitar groups BS(n,n) and BS(n,-n) automata groups?

In this article of Bartholdi and Sunik http://arxiv.org/abs/math/0603032, they say that BS(n,n) and BS(n,-n) are automata groups because they are virtually $F_{|n|}\rtimes\mathbb{Z}$ (where $F_{|n|}$ ...
2
votes
2answers
355 views

An approximate version of $g^2 = e$ for all $g \in G$, implies $G$ is Abelian [closed]

I am trying to understand the concept of approximate group. So I took a group theory exercise from a physics class at Caltech. The question basically states: Suppose that for any element $g ...
6
votes
1answer
341 views

Generators for SL_2(R) for rings of integers R

Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it? ...
15
votes
2answers
849 views

Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups. Let $p$ be a prime, and let $\mathbb{F}_p$ be the field ...
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0answers
115 views

transitive action on finite abelian subgroups [migrated]

Let G be a group and K a finite subgroup of G. Let H be some subgroup of the normalizer of K in G, and assume the action of H on K by conjugation is transitive on elements of K of same order. Does H ...
8
votes
2answers
499 views

Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
2
votes
1answer
176 views

Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ ...
6
votes
1answer
283 views

Some questions about the Malcev completion

Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n ...
2
votes
2answers
230 views

Commutator width of soluble Baumslag Solitar groups

Do the soluble Baumslag-Solitar groups have finite commutator width? A soluble Baumslag-Solitar group is given by a presentation of the from BS(1,m) = $<a,b \mbox{ }| \mbox{ } a^{-1}ba = ...
3
votes
1answer
257 views

Finitness of the Burnside Group

This is something I discussed with Andrezj Zuk, but we didn't arrive to any conclusions. Let $B(d,n)$ be the Burnside group on $d$ generators of exponent $n$. Is there an algorithm to determined ...
12
votes
1answer
265 views

Applications of Lubotzky's linearity theorem?

Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
3
votes
1answer
250 views

Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$, the group of invertible upper triangular $n\times n$ matrices. I know that if $\rho : G\rightarrow T(n,k)$ is faithful (i.e. into) then ...
2
votes
1answer
193 views

Any representation is a sub representation of direct sum of regular representation

I need a reference for the following statement: Let G be a linear algebraic group over algebraically closed field k. Let V be a finite dimensional G-module. The V is sub representation of k[G]^n for ...
8
votes
1answer
316 views

Three involutions on the set of 6-box Young diagrams

The set of $n$-box Young diagrams classifies both conjugacy classes in $S_n$ and equivalence classes of irreducible representations of $S_n$. There is an outer automorphism of $S_6$, of order 2. ...
3
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0answers
97 views

do there exist finite simple characteristic quotients of the free group of rank 2?

Let $F_2$ be the free group of rank 2. Let $Aut^+(F_2)$ be the subgroup of $Aut(F_2)$ consisting of automorphisms of determinant 1 under abelianization. Do there exist maximal normal finite index ...
40
votes
1answer
992 views

Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
0
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0answers
47 views
2
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1answer
99 views

Transitive actions of $Aut(F_2)$ on surjections from $F_2\twoheadrightarrow G$

Here $F_2$ is the free group on two generators $x,y$. I'm interested in examples of finite groups $G$ such that $Aut(F_2)$ acts transitively on the set of surjections $F_2\rightarrow G$. (In ...
11
votes
3answers
665 views

Your favorite papers on geometric group theory

I would like to improve my culture on geometric group theory, so I am interested in short and accessible papers on subjects related to this field but which are not always available in the classical ...
14
votes
3answers
3k views

Groups with all subgroups normal

Is there any sort of classification of (say finite) groups with the property that every subgroup is normal? Of course, any abelian group has this property, but the quaternions show commutativity ...
5
votes
1answer
372 views

Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?

I not really familiar with these subjects. I read this question and I was really surprised by the answer. My question is probably vague (so please do bear with me). The cited question/answer ...
2
votes
2answers
127 views

Detecting HNN-Extension and free products with amalgamation

This question is partly connected with the following Connection between Stalling's end theorem and Seifert-van Kampen Theorem. By Stalling's Theorem a group with more than one end splits over a ...
2
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1answer
165 views

How bad can an infinite linear torsion group be?

Following this question I wonder about the following. Examples of infinite torsion groups which are linear in zero characteristic are infinite groups of roots of unit. Are there other examples ...
3
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1answer
85 views

Classification of finite abelian hypergroups and table algebras

Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...
-2
votes
1answer
70 views

On Sylow subgroup of a finite group [closed]

Let $p\mid n$, then by $n_p$ we mean the $p$-part of $n$, i.e. $n_p = p^k$ if $p^k\mid n$ but $p^{k+1}\nmid n$. Let $G$ be a finite group, $M\leq G$ and $P\in Syl_p(G)$. Is It true that $|M\cap ...
3
votes
1answer
159 views

variance of the number of fixed points for a permutation group

It is reasonably well-known that the variance of the number of fixed points for $S_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I ...
3
votes
1answer
124 views

Connection between Stalling's end theorem and Seifert-van Kampen Theorem

Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
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0answers
70 views

Obtaining a quasi-isometry of the 'boundary'

It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...
21
votes
6answers
2k views

Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ...
3
votes
1answer
223 views

Infinite groups of finite exponent inside of SL(2,C)

Fix an integer $n>0$. Are there infinite subgroups of $SL_2(\mathbb{C})$ such that every element is $n$-torsion?
23
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1answer
856 views

A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with ...
6
votes
2answers
329 views

How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple. I could not think of a better title, so let me explain my question in more detail. I have a number field $E/\mathbb{Q}$, ...
14
votes
2answers
1k views

Non-degenerate alternating bilinear form on a finite abelian group

I asked this question on math.stackexchange yesterday, but nobody has helped so far, and only 44 people have seen it! So I hope people do not mind me asking it here... Let $A$ be a finite abelian ...
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0answers
97 views

Maximizing sum of matrices

Over the last few months, I've been trying to find the solution to a research-related problem I'm having. However, my research is not in mathematics, and my progress toward reaching a solution has ...
0
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0answers
77 views

Coxeter Subgroups of Coxeter Groups

Is there an algorithm to determine all the Coxeter subgroups of a given Coxeter group? If we only want the Coxeter subgroups of finite index does that make the question easier? If we only want a ...