Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,914
questions
2
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Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
4
votes
1
answer
256
views
Subgroup of p-adic units
Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
2
votes
0
answers
99
views
Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
1
vote
0
answers
57
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Finitely presentable groups are residually finite if and only if they are universally pseudofinite
Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
9
votes
1
answer
299
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Is the group of translations of an affine plane always commutative?
$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
1
vote
0
answers
131
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Which groups can be generated by a single conjugacy class?
How can we characterize the finite groups generated by a subset of a single conjugacy class?
This post asks for well-known families of finitely generated groups generated by a single conjugacy class. ...
1
vote
0
answers
121
views
Bounds for the orders of second largest subgroups of $\mathrm{SL}_n(\mathbb F_q)$
$\DeclareMathOperator\SL{SL}$By Patton's thesis, except for a finite number of possibilities, the $(n-1, 1)$ parabolic subgroup, $P$ say, has the largest number of elements among all non-trivial ...
3
votes
1
answer
176
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Finite-maximal subgroups of orthogonal groups
I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite.
My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
4
votes
2
answers
203
views
Order of abelian subgroup of the automorphism group of an abelian group
Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
2
votes
0
answers
70
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Implementation of the nerve of a category in GAP
I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...
2
votes
0
answers
218
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Interpretation of Kazhdan T property cohomologically
$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology.
In general, we heuristically have $H^1(G,Ad(V))$ (...
3
votes
1
answer
270
views
Number of conjugacy classes of pairs of commuting elements
Let $G$ be a finite group and denote by $r_G$ the number of conjugacy classes of pairs of commuting elements, i.e. the cardinality of the following set $$ A_G = \{ c(a_1,a_2) \ | \ a_1,a_2 \in G \text{...
11
votes
2
answers
669
views
How small can maximal subgroups be?
Given a finite group $G$, let $p(G)$ denote the number of prime factors
of the order of $G$ (counting multiplicities).
Does there exist a function $f: \mathbb{N} \rightarrow \mathbb{N}$
which grows ...
3
votes
0
answers
100
views
Finite approximability of graphs with finitely many automorphisms
In this question, all graphs are understood to be simple and undirected, and have countably many vertices and edges, but not necessarily finite.
Let $G = (V, E)$ be a graph. It is clear that any ...
2
votes
1
answer
153
views
Order of a loop around a cone point
Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
2
votes
0
answers
86
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
1
vote
0
answers
127
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Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
0
votes
0
answers
49
views
Approximating open subset of profinite group by union of cosets of ideal
I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
4
votes
1
answer
126
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Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
-1
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0
answers
38
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What's a (single-sorted) algebraic signature? [migrated]
I am participating in an undergrad project that uses the book Nominal Sets by Andrew M. Pitts. I don't fully understand half of the things he attempts to explain but that is another issue, the main ...
4
votes
1
answer
286
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Extending primitive systems in free groups
It is a version of a question I asked on Math.StackExchange (no answers yet). Suppose $a_1,a_2,\ldots,a_n,b$ is a primitive system in a non-abelian free group $F$ (a part of a basis of $F$). Let $c \...
7
votes
0
answers
167
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Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
3
votes
0
answers
134
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A question about Gromov's proof of a "more effective version of the main theorem"
In the paper "Groups of polynomial growth and expanding maps" Gromov proves the following "effective version of the main theorem"
For any positive integers $d$ and $k$, there ...
2
votes
0
answers
79
views
Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?
Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
6
votes
1
answer
355
views
Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
4
votes
0
answers
175
views
Polynomials of growth for finite Heisenberg groups
Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...
6
votes
2
answers
381
views
Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$
$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$.
Now, let $n$ be an integer larger than $2$.
Question: In which circumstances, $...
2
votes
2
answers
200
views
Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$
Let $S_n$ be the symmetric group on $n$ letters. When (and how) can an $n$-cycle in $S_n$ be factored into a product $xy$, where $x,y$ have orders 2,3 respectively?
More precisely, I'd like to ...
4
votes
0
answers
376
views
Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
0
votes
0
answers
34
views
Growth of cocycles in higher degrees
Let $G$ be a group with finite symmetric generating set $S$ and
let $\pi:G\rightarrow\mathcal{U}(\mathcal{H})$ be a unitary representation
of $G$ on a Hilbert space $\mathcal{H}$. A 1-cocycle with ...
9
votes
3
answers
302
views
$G$-module structure of the relation module for a presentation of a finite group $G$
Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
7
votes
0
answers
355
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
0
votes
1
answer
162
views
Reflections on subspaces of $\text{codim} > 1$
Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
0
votes
0
answers
88
views
Isomorphism in division algebras
Let $D$ be a division algebra with center $F$ and $D'$ a division algebra with center $K$, where $K$ is a Galois field extension over $F$. Let $\phi: D \otimes K \rightarrow D'$ be $K$ algebra ...
0
votes
0
answers
147
views
Residual finiteness of semidirect product $\mathbb{Z}^2\ltimes \mathbb{Z}[1/10]$ of abelian groups
Let $\mathbb{Z}[1/10]$ be an abelian group by addition. Let $\mathbb{Z}^2$ act on it by automorphisms by $x\mapsto 2x$ and $x\mapsto 5x$. Is the corresponding semidirect product $\mathbb{Z}^2\ltimes \...
5
votes
2
answers
196
views
What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What are the Schur indices of the irreps of $\SL(2,p)$? ($p$ an odd prime.)
Presumably this is in a book somewhere? Section 6 of the paper &...
1
vote
0
answers
155
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Applications of Artin's theorem on induced representations
Let $G$ be a finite group and let $R(G)$ be the (complex) representation ring of $G$. As stated in Serre's book on representation theory, Artin's theorem says the following:
Theorem: Let $X$ be a ...
0
votes
1
answer
182
views
Fixed points free automorphisms of Teichmüller spaces
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
6
votes
1
answer
400
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Relations between relations in the positive braid monoid
The positive braid monoid on $n$ strands is the monoid with generators $s_1$, $s_2$, ..., $s_{n-1}$ and relations
$$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \qquad s_i s_j = s_j s_i \text{for}\ |i-j| \...
4
votes
3
answers
547
views
Regular orbits for automorphisms of finite simple groups
Let $G$ be a finite group and $f$ be an automorphism of $G$. We say that $f$ has a regular orbit if there exists $x\in G$ such that $|x^f|=|f|$. If $G$ is abelian it is known that every automorphism ...
4
votes
0
answers
190
views
Infinite groups with 2 automorphism orbits
A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(...
14
votes
0
answers
469
views
Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
7
votes
1
answer
604
views
What are double groups mathematically?
In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" ...
0
votes
1
answer
166
views
Are all "almost projective" groups free?
Let us say that a group $H$ is almost projective if, given any group epimorphism $f\colon G\to H$, there is an embedding $i\colon H\to G$.
Does it follow that $H$ is free? If not, is there a ...
9
votes
1
answer
500
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Shortest almost trivial element of free group [duplicate]
Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$.
Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.
What is the ...
9
votes
1
answer
365
views
Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $F_2$ be a free group of rank 2. There is a surjection $\Aut(F_2)\rightarrow \GL(2,...
2
votes
1
answer
220
views
An interior cone condition for Teichmuller spaces
Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
3
votes
1
answer
218
views
Nonisomorphic central products on the same pair of groups?
A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such ...
2
votes
0
answers
91
views
What is the complexity / name of word search problem in linear groups?
This is a question about a search problem associated with user6976's question. Suppose we are given a finite set of elements $S \subset \mathrm{GL}_n(\mathbb{Q})$ containing inverses of all its ...
3
votes
1
answer
259
views
Distinct characters with the same character values, outer automorphisms and Galois conjugation
Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:
multiplying by a degree 1 character
applying an ...