All Questions
8,438 questions
0
votes
1
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73
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Can we construct an isomorphism between $\mathrm{BS}(1,n)$ and $\mathbb{Z}[1/n]\rtimes\mathbb{Z}$ such that it preserve the order?
It is given in Regular left-orders on groups that the solvable Baumslag-Solitar group $\mathrm{BS}(1,n)=\langle a, b\mid aba^{-1}=b^n\rangle $ is isomorphic to $\mathbb{Z}[1/n]\rtimes \mathbb{Z}$ for ...
-4
votes
1
answer
163
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What are all the complex structures on $\mathbb{R}^2$ which live inside $\mathrm{SL}_2(\mathbb{Z})$? [closed]
By "complex structure" I am referring to 2x2 matrices which square to $-\mathrm{Id}_2$. I need to know those with integer entries and determinant equal to 1.
Thank you
- 3
5
votes
1
answer
238
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Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as
$$\pi_1(G, T):...
- 53
13
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4
answers
1k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
-3
votes
0
answers
156
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A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
2
votes
0
answers
63
views
Classification of centralizers of elements of finite simple groups of Lie type
I am currently studying the twisted Ree finite simple groups given by $^2G_2(3^{2n+1})$ and I was wondering if there is a reference for the classification of centralizers of elements in this family of ...
- 71
3
votes
0
answers
157
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Faithful representations and symmetric powers
In the question Faithful representations and tensor powers, several proofs demonstrate that for every faithful complex representation $V$ of a finite group $G$, every irreducible complex ...
1
vote
1
answer
266
views
Group element of group algebra
For a prime $p$, let $G$ be a finite $p$-group and $F_{p}$ the field with $p$ elements.
Let $A=\{a\in F_{p}G \mid a^{\sum_{x\in G}x}\neq 0\}$, where $F_pG$ is the group algebra of $G$ over $F_p$ and $...
- 171
0
votes
0
answers
72
views
Co-boundary crossed homomorphism & "sign" preserving. Why 2-valued components is special?
Suppose $h_{g}: \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a coboundary crossed homomorphism with action $g$ as a cyclic permutation of coordinates on $\mathbb{R}^n$ vectors. So, the acting group is a ...
- 11
5
votes
0
answers
95
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$\text{Rep}(D_4)$ and its three fiber functors
It is well-known that the fusion category $\text{Rep}(D_4)$ of representations of the dihedral group $D_4$ of order 8 admits three distinct fiber functors. Therefore, there are three different Hopf ...
8
votes
0
answers
201
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Logarithm of a $p$-group in $\mathrm{GL}_n(p)$
$\def\GL{\operatorname{GL}}\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}\def\Id{\mathrm{Id}}\def\fu{\mathfrak{u}}$Let $p$ be prime, let $n<p$, let $U_n(\FF_p)$ be the group of $n \times n$ upper ...
- 156k
2
votes
0
answers
80
views
Question about lattice with dense projection
Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
- 305
3
votes
1
answer
343
views
Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$
The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
- 48.9k
3
votes
1
answer
160
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
12
votes
0
answers
342
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
4
votes
2
answers
207
views
Normalizers of the principal congruence subgroups in $\mathrm{GL}(n,\mathbf Q)$
A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence ...
- 43
0
votes
0
answers
136
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What are the automorphisms of finite commutative groups？ [migrated]
What are the automorphisms of finite commutative groups？Is there a relatively complete conclusion? Although it can be decomposed into the direct product of cyclic groups, this question still seems ...
- 83
1
vote
0
answers
44
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Lower bound for restricted sumset in ordered groups
Recently in The restricted sumsets in finite abelian groups it is proved that
Suppose that $k \geq 2$ and $A$ is a non-empty subset of a finite abelian
group $G$ with $|G| > 1$. Then the ...
-3
votes
2
answers
195
views
Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
- 83
0
votes
1
answer
142
views
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
- 83
1
vote
1
answer
187
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Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
2
votes
0
answers
164
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Nonabelian groups where every element has small order
Let $G$ be a finite nonabelian group with the property that if $g \in G$, then
$$\DeclareMathOperator{\ord}{ord} \ord(g) \leqslant 10 \log_2 |G|, $$
where $\ord(g)$ is the order of the element $g$, ...
- 1,354
2
votes
2
answers
206
views
Software library for complex irreducible representations of $\mathrm{PSL}_n(q)$
I came across an extremely useful Python software library for the Monster group: https://github.com/Martin-Seysen/mmgroup which allows for all sorts of manipulations involving the sporadic finite ...
- 1,150
2
votes
1
answer
153
views
On generation of $A_n$ by elements of prime order
There is a question regarding generation of finite simple groups with elements of prime order. Recently, Guralnick, Shareshian, Woodroofe and Teräväinen made advances in this direction. We have, for ...
- 329
7
votes
1
answer
289
views
Group cohomology valued in a bimodule
The usual setup for group cohomology of a group $G$ is as follows. One takes a $G$-module $M$, and considers the space of all maps
$$\ell : G \times \cdots \times G \longrightarrow M $$
together with ...
- 9,853
5
votes
1
answer
367
views
Groups with no proper non-trivial fully invariant subgroup
Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be characteristic if $\phi(H)\subseteq H$, $\forall \phi \in \operatorname{Aut}(G)$ and fully invariant if $\phi(H)\subseteq H$, $\forall \...
- 361
2
votes
0
answers
102
views
Finite groups of Lie type
Table $22.1$ Finite groups of Lie type from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
For the type $A$: $G_{sc}^{F}=\operatorname{SL}_{n}(q)$ and $G_{ad}...
- 217
1
vote
1
answer
80
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$p$-torsion related to algebraic groups
Definition $14.14$ from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
A prime $p$ is a torsion prime for a linear algebraic group $G$ if the fundamental ...
- 217
19
votes
0
answers
478
views
On C*-rigidity problem for torsion-free groups
I'd like to address the $\mathrm{C}^\ast$-rigidity problem for
torsion-free groups (see
this paper),
which asks for non-isomorphic torsion-free groups with isomorphic
(reduced) group $\mathrm{C}^\ast$-...
- 10.1k
4
votes
1
answer
378
views
Where to begin in Computational Group Theory?
I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
- 297
2
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0
answers
165
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Definition for "almost simple" linear algebraic groups
Proposition 2.18 from "Elementary abelian $p$-subgroups of algebraic groups" by R. Griess. used the term "simply connected almost simple linear algebraic group $G$" without ...
- 217
1
vote
0
answers
72
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Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
0
votes
0
answers
95
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Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
8
votes
1
answer
1k
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GAP cannot solve Rubik's cube 4x4x4 and higher ? (Practical limits of Schreier–Sims algorithm)
According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. That means cannot decompose given random element of ...
- 24.9k
16
votes
1
answer
977
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
0
votes
1
answer
64
views
Transitive map on a profinite group
Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
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0
votes
0
answers
61
views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
7
votes
1
answer
310
views
Homotopy between posets
This is entirely a new area for me and I apologise in advance if the questions are silly.
In Quillen's paper "Homotopy properties of the posets of non-trivial $p$-subgroup of a group" (see ...
- 139
12
votes
1
answer
323
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Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
The title is pretty much the whole question. Let $S_g$ be a closed, oriented surface of genus $g$. Does there exist $n$ such that the mapping class group $\mathrm{Mod}(S_g)$ embeds as a subgroup of $\...
- 3,740
4
votes
1
answer
161
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Finite group is Dedekind iff for every irrep, every element acts as identity or has all eigenvalues $\ne 1$
Consider the following claim: a finite group $G$ is Dedekind $\iff$ for every irrep $\rho$, and every $g \in G$, $\rho(g)$ either is identity matrix or has all eigenvalues $\ne 1$.
Is this claim true?
...
- 393
1
vote
0
answers
100
views
Lawvere theory and presentations of groups
In his dissertation on "Functorial semantics of algebraic theories", Lawvere says in his introduction that
"from the category (or more precisely from an underlying-set functor)
we can ...
2
votes
1
answer
161
views
Smallest dimensional faithful complex representation of $\mathrm{PSL}(k,q)$
For given $k>1$ and $q$ a prime power, what is the minimal dimension, as a function of $(k,q)$, for which a faithful complex representation of the projective special linear group over $\mathbb{F}_q$...
- 1,150
5
votes
0
answers
543
views
A problem on additive combinatorics in right-ordered groups
In a paper Small doubling in ordered groups: generators and structure it is proven in Lemma 4 page no. 598 that:
Let $G$ be an ordered group. Let $S$ be a finite subset of $G$ with at least two ...
4
votes
0
answers
115
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Complexity to find "short" (e.g. polynomial in diameter) decomposition of the permutation into the product of generators?
Question 1: Consider the symmetric group $S_n$ and some set of permutations $p_i$. Given permutation $g$ - what is known about the algorithmic complexity to decompose $g$ into product of $p_i$ ...
- 24.9k
9
votes
0
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292
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Tilings in finite (not necessarily Abelian) groups
Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that
$$ G = \bigsqcup_{b\in B} bA.$$
...
- 1,354
4
votes
1
answer
441
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Large(ish) finite non-abelian subgroups of $\operatorname{GL}_n \mathbb C$ for $n>70$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an ...
- 1,150
16
votes
2
answers
602
views
$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 911
0
votes
1
answer
99
views
Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
- 361
9
votes
1
answer
304
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About the normal subgroups of Burnside groups
I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
- 91
3
votes
0
answers
89
views
Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
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