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

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33
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1k views

Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$. ...
25
votes
0answers
413 views

Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid ...
21
votes
0answers
766 views

Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2, $$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
20
votes
0answers
882 views

Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory (Contains some quotes from Maciej Malicki's paper.) The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...
20
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0answers
572 views

Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
19
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0answers
681 views

Non-linear expanders?

Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...
19
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548 views

Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
18
votes
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425 views

Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon. Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...
18
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0answers
346 views

Characteristic subgroups and direct powers

Solved question: Suppose H is a characteristic subgroup of a group G. Is it then necessary that, for every natural number n, in the group $G^n$ (the external direct product of $G$ with itself $n$ ...
17
votes
0answers
686 views

Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...
16
votes
0answers
354 views

Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?

Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later ...
16
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577 views

Quotients of residually finite groups by amenable normal subgroups

My questions are: Is there any group, which cannot be written as the quotient of a residually finite group by an amenable normal subgroup? Is it possible for large classes of groups? and Is ...
15
votes
0answers
268 views

Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction. Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
15
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0answers
495 views

Are amenable groups topologizable?

I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
14
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259 views

Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series $\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely. Then there is a partition of the symmetric group ${\rm ...
14
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0answers
504 views

What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11, London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
14
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689 views

Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?

Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable. ...
14
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757 views

Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split

Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$ Actually this ...
14
votes
0answers
438 views

The multiplication game on the free group

Fix $W\subseteq\mathbb F_2$ and consider the following two-person game: Player 1 and Player 2 simultaneously choose $x$ and $y$ in $\mathbb F_2$. The first player wins, say one dollar, iff $xy\in W$. ...
14
votes
0answers
483 views

Motivation for Hall-Witt identity

I've wondered for a while about the (Hall-)Witt identity in group theory: $[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}]],b]^a = 1$. (Here, $x^y$ means $y^{-1}xy$ and $[x,y]$ ...
14
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0answers
580 views

Galois theory: Generalization of Abel's Theorem?

Let $L$ stand for the field obtained by adjoining to ${\Bbb Q}$ all roots of all polynomials of the form $x^n+ax+b$, $a,b\in {\Bbb Q}$. What polynomials $p$ don't split over $L$? In particular, ...
13
votes
0answers
201 views

Bounding the lengths of the conjugators in the word problem for finite group presentations

Let $G = \langle X \mid R \rangle$ be a group defined by a finite presentation, and let $F$ be the free group on $X$. If $w \in F$ represents the identity in $G$, then $w$ is equal in $F$ to (the free ...
13
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268 views

Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE. There exists a rather remarkable ...
13
votes
0answers
613 views

The symmetric group and the field with one element

I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of ...
13
votes
0answers
474 views

Some questions on unitarisability of discrete groups

In this post I would like to ask several of questions related to Dixmier problem. I will try to make the post as self-contained as possible. A discrete group $G$ is unitarisable if for every Hilbert ...
12
votes
0answers
238 views

Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero. In the 1966 paper "Additive gruppen mit vorgegebener ...
12
votes
0answers
294 views

What is an example of a word hyperbolic group without a finite complete rewriting system?

I believe that it was an open question back when I was a graduate student whether every word hyperbolic group admits a finite complete (=Church-Rosser=Noetherian+confluent) rewriting system for some ...
12
votes
0answers
319 views

Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but: Question: Is there a reformulation of the Dynamic ...
12
votes
0answers
292 views

Nilpotent pro-$p$ groups

Is it true that every finitely generated (topologically) torsion free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
12
votes
0answers
576 views

What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...
12
votes
0answers
572 views

Are all these groups CAT(0) groups?

Given a geodesic metric space $X$ together with a choice of midpoints $m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$). Assume furthermore, that the following nonpositive ...
12
votes
0answers
431 views

A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as $$ ...
11
votes
0answers
155 views

When is a group Fibonacci sequence contained in a single conjugacy class?

First a definition: a Fibonacci sequence in a group is a sequence in which the first two elements may be arbitrary, and from there on each element is a product (using the group operation) of the ...
11
votes
0answers
278 views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability ...
11
votes
0answers
348 views

Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?

(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...
11
votes
0answers
155 views

Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
11
votes
0answers
400 views

Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers

Definition / Question Definition: 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 ...
11
votes
0answers
522 views

Groups generated by 3 involutions

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 ...
10
votes
0answers
306 views

$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...
10
votes
0answers
232 views

Groups with reduced C*-algebras of stable rank 1

Let $G$ be a countable discrete group, $C_r^*(G)$ its reduced $C^*$-algebra. We say that $G$ has stable rank 1 if $C_r^*(G)$ has stable rank one, that is, the set of invertible elements is dense in ...
10
votes
0answers
132 views

Star-shaped Folner sequence

Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that ...
10
votes
0answers
283 views

Characteristic subgroup of nilpotent group that is not invariant under powering

I want an example of a nilpotent group $G$, a characteristic subgroup $H$, and a prime number $p$ such that: $G$ is $p$-powered, i.e., every element of $G$ has a unique $p^{th}$ root in $G$. $H$ is ...
10
votes
0answers
325 views

Group rings isomorphic over F_p, but not over Z_p ?

Suppose given a prime $p$. Question: Do there exist finite groups $G$ and $H$ such that ${\bf F}_p G$ is isomorphic to ${\bf F}_p H$, but such that ${\bf Z}_p G$ is not isomorphic to ${\bf Z}_p H$ ? ...
10
votes
0answers
484 views

Centralizers of group actions

Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes ...
10
votes
0answers
349 views

Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?

Let $R$ be a commutative ring, and, for $n\ge0$, ${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series $u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which $a_0\in R^\times$ and $u(x)\equiv ...
10
votes
0answers
376 views

Where can I find analogues of combinatorial central limit theorems for other groups

The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each $n$, an $n \times n$ matrix $A = (a_{ij})$, one can consider the random diagonal sum: $$\displaystyle ...
9
votes
0answers
121 views

Elements of order 3 normalizing no non-identity 2-subgroups in Almost Simple Groups

This question is partly motivated by a situation which arises in modular representation theory. A finite group $G$ is said to be almost simple if $G$ has a unique minimal normal subgroup which is a ...
9
votes
0answers
113 views

Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent). Consider an element ...
9
votes
0answers
256 views

Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$ Let $A$ be a finite abelian group and $G$ a finite group acting on $A$. Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
9
votes
0answers
418 views

How many sporadic simple groups are there, really?

I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...