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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 ...
SPDR's user avatar
  • 103
2 votes
0 answers
55 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 ...
MAP's user avatar
  • 71
3 votes
0 answers
154 views

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 ...
LuckyJollyMoments's user avatar
1 vote
1 answer
255 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 $...
gdre's user avatar
  • 171
0 votes
0 answers
58 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 ...
Nartoo Meon's user avatar
5 votes
0 answers
88 views

$\text{Rep}(D_4)$ 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 ...
Alonso Perez-Lona's user avatar
8 votes
0 answers
196 views

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 ...
David E Speyer's user avatar
2 votes
0 answers
76 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 ...
studiosus's user avatar
  • 305
3 votes
1 answer
340 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 $\...
Dominic van der Zypen's user avatar
3 votes
1 answer
153 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 $\...
user82261's user avatar
  • 357
12 votes
0 answers
328 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 $\operatorname{PGL}_d(R)$? ...
Carl Schildkraut's user avatar
4 votes
2 answers
204 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 ...
P.H.'s user avatar
  • 43
0 votes
0 answers
136 views

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 ...
user530909's user avatar
1 vote
0 answers
44 views

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 ...
navashree chanania's user avatar
-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.
user530909's user avatar
0 votes
1 answer
141 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.
user530909's user avatar
1 vote
1 answer
186 views

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 ...
Maziar Esfahanian's user avatar
2 votes
0 answers
162 views

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 $g$, ...
Anurag Sahay's user avatar
  • 1,354
2 votes
2 answers
205 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 ...
Fetchinson0234's user avatar
2 votes
1 answer
152 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 ...
Lucas's user avatar
  • 329
7 votes
1 answer
288 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 ...
Matthias Ludewig's user avatar
5 votes
1 answer
362 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 \...
Nick Belane's user avatar
2 votes
0 answers
99 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}...
scsnm's user avatar
  • 217
1 vote
1 answer
79 views

$p$-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 ...
scsnm's user avatar
  • 217
19 votes
0 answers
472 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 $\mathrm{C}^\ast$-...
Narutaka OZAWA's user avatar
4 votes
1 answer
375 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 ...
G. Fougeron's user avatar
2 votes
0 answers
163 views

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 ...
scsnm's user avatar
  • 217
1 vote
0 answers
71 views

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 $...
Zhiyu's user avatar
  • 6,582
0 votes
0 answers
93 views

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_{\...
user545662's user avatar
8 votes
1 answer
1k views

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 ...
Alexander Chervov's user avatar
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 $\mu(H)=0$ ...
Nick Belane's user avatar
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 $S_n$. ...
dbossaller's user avatar
7 votes
1 answer
308 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 ...
GURI920826's user avatar
11 votes
1 answer
320 views

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 $\...
Matt Zaremsky's user avatar
4 votes
1 answer
159 views

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? ...
Manu's user avatar
  • 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 ...
Sylvain Cabanacq's user avatar
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 $\mathbb{F}_q$...
Fetchinson0234's user avatar
5 votes
0 answers
538 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 ...
navashree chanania's user avatar
4 votes
0 answers
115 views

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 $p_i$ ...
Alexander Chervov's user avatar
9 votes
0 answers
291 views

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 $$ G = \bigsqcup_{b\in B} bA.$$ ...
Anurag Sahay's user avatar
  • 1,354
4 votes
1 answer
440 views

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 ...
Fetchinson0234's user avatar
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 ...
Marcos's user avatar
  • 911
0 votes
1 answer
98 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 $G$...
Nick Belane's user avatar
9 votes
1 answer
304 views

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 ...
GroupKing's user avatar
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 ...
YC Su's user avatar
  • 605
14 votes
2 answers
742 views

Solving the Bring quintic using the Monster?

I. Method Hermite's method to solve the Bring quintic by functions that obey $x^8+y^8=1$ implicitly uses octahedral symmetry, while Emil Jann Fiedler's solution by the Rogers-Ramanujan continued ...
Tito Piezas III's user avatar
2 votes
0 answers
166 views

Centralizer of PSL in PGL and of SL in GL: reference request

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
Nick Belane's user avatar
9 votes
1 answer
1k views

Is the number of varieties of groups still unknown?

A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
Martin Brandenburg's user avatar
3 votes
1 answer
181 views

In dimension $n=5$, does a subgroup of $O(n)$ satisfying these properties exist?

I asked a question where @YCor provided a construction that seems to enable a group construction satisfying some properties when $n\ne 5$. However, in the case $n=5$, I am starting to think no such ...
Quoka's user avatar
  • 185
4 votes
0 answers
97 views

Characterization of Vilenkin group

It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
John's user avatar
  • 85

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