Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
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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
$...
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Connes & Marcolli: Q-lattices generalize Conway's "Understanding groups like $\Gamma_0(N)$"
Has anyone generalized Conway's description of Hecke operators on lattices to the Q-lattices of Connes & Marcolli ?
Light may well be shone on moonshine thus.
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Commutator subgroup of a surface group
Let $\Sigma_{g,n}$ denote a compact orientable genus $g$ surface with $n$ boundary components. Assume that $g \geq 1$ and fix a basepoint $p \in \Sigma_{g,n}$. Define $S \subset [\pi_1(\Sigma_{g,n},...
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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
$$
v_{m,...
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On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
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God's number for higher dimensional Rubik's cubes
In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every ...
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Minimal actions commuting with amenable actions of $\mathbb{F}_2$
For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...
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Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?
Let $G$ be a finite group. It has been shown that:
If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
If the probability ...
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Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$
Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
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Roadmap to homotopical group theory
I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the ...
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Isomorphic free groups have bijective generating sets
Let $F(X)$ be the free group on a set $X$. Classically, we can prove the statement:
$F(X) \cong F(Y)$ if and only if $|X|=|Y|$.
The proofs (that I have seen) consist of turning the group ...
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Ascending chain condition for 1-element normal closures in a free group
Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?
In other words, can there exist elements $...
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Proving a group with two generators is not free that uses the Brahamagupta-Pell equation
Hello I encountered the following while reading a set of notes on free groups. It's not a homework question.
"Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
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Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
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If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?
It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle?
...
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Cyclic and prime factorizations of finite groups
A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$.
In Cryptology factorizations of groups are known as ...
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Modular version of Mednykh's formula?
Let $G$ be a finite group and $\Sigma_g$ a closed Riemann surface of genus $g$. Then Mednykh's formula states
$\frac{\left|\mathrm{Hom}(\pi_1(\Sigma_g),G)\right|}{\left|G\right|} = \frac{1}{\left|G\...
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When does the first subgroup growth function grow?
Given a group $G$, the subgroup growth functions are given by
$a_n(G) :=$ the number of subgroups of $G$ of index exactly $n$.
$s_n(G) :=$ the number of subgroups of $G$ of index at most $n$.
The ...
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Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
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Iterated automorphism groups of finite groups
Let $\mathcal{G}$ be the set of isomorphism classes of finite groups.
There is an operation $\mathrm{Aut} : \mathcal{G} \rightarrow \mathcal{G}$ which gives the automorphism group of a given group, ...
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Is it decidable if a tree-presented semigroup contains an idempotent?
A semigroup presentation $\langle A | R\rangle$ is called tree-like if every relation has the form $ab=c$, $a,b,c$ are in $A$ and if two relations $ab=c, a'b'=c'$ belong to $R$, then $c=c'$ if and ...
user6976
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Sequence of proper retracting homomorphisms between finitely presented groups?
Let $G$ be a group. Recall that a group $H$ is called a retract of $G$ if there exist homomorphisms $g:G\longrightarrow{H}$ and $f:H\longrightarrow G$ so that $g\circ f=id_H$. The homomorphism $g$ ...
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Other than the Higman group, what other candidates do we have for non-sofic groups?
I know that the Higman group is the most widely studied candidate right now, but what are the others? For example, is (are) Thompson's group(s) sofic? And what about the Burger-Mozes groups? I haven't ...
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Detecting a module for the free group algebra on a finite quotient
Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...
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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 ...
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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 ...
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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 $...
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Proving amenability of an extension by using paradoxical decompositions
It is well known that an extension of an amenable group by an amenable group is amenable. Is it possible to prove that by using only paradoxical decompositions: if $G$ has a paradoxical decomposition ...
user6976
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Topology of marked groups for different number of generators
A $k$-marked groups is a pair $(G,S)$ where $G$ is a group and $S$ is an ordered set of $k$ generators of $G$. Each such pair can be identified with a normal subgroup of the free group $F_k$ of rank $...
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Uniquely geodesic groups
Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...
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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 ...
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Properties of Artin, Shephard and generalized von Dyck groups
I think that the problems below are open, but maybe I missed something. Definitions follow the problems. All groups are assumed to be finitely generated.
Question 1. Is every Artin group (resp. ...
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Shortest path in Cayley graphs
The standard way to find the shortest path between 2 vertices, $v_1$ and $v_2$, of an undirected graph is BFS (breadth first search) which takes time $O(|E|)$ and space $O(|V|)$ (where $E$ is the set ...
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What do Multilinear Forms tell us about Representations?
The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
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Kissing the Monster, or $196,560$ vs. $196,883$
The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...
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the (non-existent) group of conformal transformations
In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what ...
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Twisted crossed product von Neumann Algebras
I asked a question over on Math.stackexchange a few days ago, but it didn't get much activity. Hopefully this question isn't considered too elementary by the standards of Mathoverflow. Here is what I ...
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Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?
A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width.
Let $X$ be a geodesically complete CAT(0) ...
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Surjective homomorphisms between braid groups
There cannot be a surjective homomorphism $B_2 \to B_n$ for any $n > 2$ because $B_2$ is commutative and $B_n$ is not. It seems plausible that if $m < n$, there cannot be a surjective ...
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Recognizing reflection subgroups of Coxeter groups
Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
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How to describe the power operation on Lie groups?
Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$, or its compact form over $\mathbb{R}$. Recall that the automorphism group $\operatorname{Aut}(\mathfrak{g})$ is of the form $G^{\...
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Finite quotients of surface braid groups
Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$.
There is a braid $A_{12} \in \Sigma_b$ ...
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Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
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Is there a one-relator circle-packing theorem?
Let $X_w$ be the presentation complex of a one-relator group $\langle x_1,\dotsc,x_n\mid w\rangle$, with $w$ cyclically reduced, i.e., $X_w=R\cup_w D$, with $R$ the rose with $n$ petals labeled $x_1,\...
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On the Number of Parallel Automorphism Lines
Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...
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2-generator subgroups of an Artin group of small type
Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...
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What are the analogs of a Levi/Parabolic/Borel/Bruhat over the field with 1 element?
This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway.
If I understand correctly, for any reductive algebraic group $G$ ...
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Number of matrices with 'small' entries in integer matrix groups
Let $n$ be a positive integer, and let $G \leq {\rm GL}(n,\mathbb{Z})$
be finitely generated. Given a bound $b \in \mathbb{N}$, let $e_b$ be
the number of elements of $G$ all of whose matrix entries ...
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Sets which are unions of translates of each other but aren't single translates
I'm a hobbyist mathematician so any question I ask here might be at risk of closure. I hope this one is good enough, but I'm not sure. This is a continuation of two questions I asked on math....
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Group presentations with few relations and undecidable word problem
This question came up in an algebra class I'm teaching. It's not my field and I couldn't find an answer easily, so I thought I would ask it here.
Is the fewest number of relations in a presentation ...
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