Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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Roadmap to learning the classification of finite simple groups
I want to learn the classification of finite simple groups. But it is often commented that it is a theorem spanning tens of thousands of pages of research papers. So it is quite intimidating to an ...
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Is there a way of canonically labelling permutation groups?
When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with ...
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When does $G\times G\times G$ admit a faithful group action on a set of size $|G|$?
[Edited due to YCor's comment:]
Given a finite group $G$, under what conditions does $G\times G\times G$ (the direct product of three copies of $G$) admit a faithful group action on a set of size $|G|$...
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The image of the point-pushing group in the hyperelliptic representation of the braid group
Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation
$\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$
called the "hyperelliptic representation," which ...
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Technical issue in the approach to Lie groups taken in a book
I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...
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How did "Ore's Conjecture" become a conjecture?
The narrow question here concerns the history of one development in group theory, but the broader context involves the sometimes loose use of the term "conjecture". This goes back to older work of ...
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What is the largest Laver table which has been computed?
Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table $(A_n,...
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An example of a non-amenable exact group without free subgroups.
A countable discrete group $\Gamma$ is said to be exact if it admits an amenable action on some compact space.
So clearly amenable groups are exact, but large familes of non-amenable groups are as ...
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$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class
It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...
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Probability of satisfying a word in a compact group
This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. ...
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Asymptotics of the growth rate of a group
Let $\Gamma$ be a finitely generated group of exponential growth and $gr(S)=\lim_{k\rightarrow \infty} \sqrt[k]{|B_k(S)|}$ be the growth rate of $\Gamma$ with respect to the generating set $S$. I am ...
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Non-residually finite matrix groups
By Malcev's theorem, every finitely generated linear group is residually finite (RF).
On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...
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What is the cohomological dimension of the commutator subgroup of the pure braid group?
I'm interested in computing the cohomological dimension of the commutator subgroup $[P_n,P_n]$ of the pure braid group $P_n$. I wasn't able to find a reference in the literature.
Because $[P_n,P_n]$ ...
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Generation in a group versus generation in its abelianization.
Background
I have been spending a lot of time in my research with subsets of groups that are very close to being generating sets. To make this precise:
Let $G$ be a group. If a subset $S$ of $G$ ...
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71, the Monster, and c = 24 CFTs
The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:
The minimal faithful representation has dimension $196883 = 47.59.71$
The Monster group can ...
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Generation of finite index subgroups
Related to a question by Mark Sapir (see here) and a question by Kate Juschenko (see here), let me ask the following:
Question: Let $G$ be a finitely generated group and let $\varepsilon>0$. Is ...
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Monstrous moonshine for $M_{24}$ and K3?
An important piece of Monstrous moonshine is the j-function,
$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$
In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ...
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Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
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For which $n$ is it true that all surjections $SL_2(\mathbb{Z})\rightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?
For which integers $n$ does every surjection $SL_2(\mathbb{Z})\twoheadrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?
(this is the usual kernel, ie, the subgroup of matrices ...
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A symmetric-like group and the quaternion group $Q_8$
It is well known that the symmetric group $S_n$ admits presentation with
$\{(ij) \mid i\neq j\}$ as the set of generators and the following list of relations
(in every formula distinct letters denote ...
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Does $E_8$ know $Spin(7)$?
One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...
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What finite simple groups we can obtain using octonions?
Rearranged on 2017-05-31
What I am missing is a uniform definition of finite simple groups. Especially sporadic groups are difficult to define.
Many of the finite groups are defined using machinery ...
user21230
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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 $xy\in W$. ...
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Numbers of distinct products obtained by permuting the factors
Let $n \in \mathbb{N}$. Is it true that for every $k \in \{1, \dots, n!\}$ there are
some group $G$ and pairwise distinct elements $g_1, \dots, g_n \in G$ such that the set
$\{g_{\sigma(1)} \cdot \ \...
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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 ...
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Cogroup objects
Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in ...
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What is the geometric shape of the Monster sporadic group?
Conway made the comment that the Monster group represents the symmetries of a shape in 196,883 dimensions, something like a "star you hang on a Christmas tree."
My question is, What do we know (or ...
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Atypical use of Sylow?
The typical application of Sylow's Theorem is to count subgroups. This makes it difficult to search the web for other applications, since most hits are in the context of qualifying exams.
What are ...
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Three questions on large simple groups and model theory
Yesterday, in the short course on model theory I am currently teaching, I gave the following nice application of downward Lowenheim-Skolem which I found in W. Hodges A Shorter Model Theory:
Thm: Let $...
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"Natural" generating sets for symmetric groups
The symmetric group on $n$ letters has
many sets of generators. Some of them are more natural than others, eg the
set $(i,i+1)$ of adjacent transpositions (natural with respect to the type A Weyl ...
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Sylow Subgroups
I had been looking lately at Sylow subgroups of some specific groups and it got me to wondering about why Sylow subgroups exist. I'm very familiar with the proof of the theorems (something that ...
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Primes occurring as orders of elements of a finitely presented group
Is it true that given a finitely presented group $G$, either all primes
or only finitely many of them occur as orders of elements of $G$?
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Groups for which the $n$-power map is a homomorphism
Let $\mathcal{V}_n$ be the collection of all groups satisfying $(ab)^n=a^nb^n$ for all $a,b$.
In particular $\mathcal{V}_1$ consists of all groups and $\mathcal{V}_2$ consists of all abelian groups, ...
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Motivation behind the construction of Deligne and Lusztig
If $G$ is a connected reductive group over a finite field $\mathbb{F}_q$ and $T$ is a maximal torus in $G$, the famous construction of Deligne and Lusztig (Annals of Math, 1976) associates ...
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How many elements does it take to normally generate a group?
$\DeclareMathOperator\nr{nr}\DeclareMathOperator\rank{rank}$This is a terminology question (I should probably know this, but I don't). Given a group $G$, consider the minimal cardinality $\nr(G)$ of a ...
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A new combinatorial property for the character table of a finite group?
Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character.
Consider the following combinatorial property of $\Lambda$: for ...
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Is there a non-Hopfian lacunary hyperbolic group?
The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...
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Is anything known about this braid group quotient?
Let $B_n$ be the braid group on $n$ strands. As is well known, if $\sigma_i$ is the operation of crossing the string in position $i$ over the string in position $i+1$, then the elements $\sigma_1,\...
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Easier reference for material like Diaconis's "Group representations in probability and statistics"
I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...
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The Monster Group uses in mathematical physics
I am doing a project on the inverse Galois problem, and am seeking to show that the monster group is realisable over the rationals. I have heard that the monster group has found uses in theoretical ...
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Is a reductive adelic group a Type I group?
I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...
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Abelianization of a semidirect product
I believe there is a straightforward formula for the abelianization of a semi-direct product: if $G$ acts on $H$, and we form the semi-direct product of $G$ and $H$ in the usual way, and the ...
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Geometric interpretation of group rings?
For a group $G$, is there an interpretation of $\mathbb C[G]$ as functions over some noncommutative space?
If so, what does this space "look like"? What are its properties? How are they related to ...
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size of smallest generating set of a group
Suppose I have a nice (e.g., word-hyperbolic? bi-automatic? automatic?) group and I want to know how big the smallest generating set is. Is that tractable (or, to put it more optimistically, what is ...
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"The" random tree
One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ ...
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The number of maximal subgroups up to isomorphism
Every maximal subgroup of infinite index of a free non-cyclic group $F_k$ is free of countable rank. Thus even though the set of maximal subgroups of $F_k$ is uncountable, there are only countably ...
user6976
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Is applying Feit–Thompson’s theorem for the nonexistence of a simple group of order $1004913$ really a circular argument?
In p.212 of Dummit–Foote’s Abstract Algebra, 3rd Edition, an analysis of a hypothetical simple group $G$ of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$ is carried out. The authors write:
We ...
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Groups of order $n$ with a character whose degree is at least $0.8\sqrt{n}$ (say)
[edited in response to some corrections by Geoff Robinson and F. Ladisch]
Throughout, all my groups are finite, and all my representations are over the complex numbers.
If $G$ is a group and $\chi$ ...
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Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
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Meaning of 'alternating' group ?
What's the meaning of the adjective 'alternating' in the name of the 'alternating group' ?
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