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-2 votes
0 answers
91 views

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 ...
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 ...
0 votes
1 answer
203 views

Equivalence of dihedral and symmetric group actions on a specialized real algebra

Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help. consider first the case where the digit 7 is not allowed, simplifying the ...
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 $...
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 ...
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 ...
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 ...
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 $\...
5 votes
0 answers
87 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 ...
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 ...
12 votes
0 answers
325 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 $\operatorname{PGL}_d(R)$? ...
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 $\...
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 ...
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 ...
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 ...
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 ...
-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.
4 votes
0 answers
603 views

Show me that I have not simplified the proof of the Adian-Rabin theorem

Let $G$ be a group with presentation $\langle x_1,x_2...,x_m|R \rangle$ and let $G'=G \ast \langle y_0 \rangle$. Now define $y_i=y_{0}x_i$. Notice that $G'=\langle y_0,y_1,...y_m\mid R'\rangle$ for ...
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.
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 ...
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 ...
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 $g$, ...
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 ...
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 ...
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 \...
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}...
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 $\mathrm{C}^\ast$-...
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 ...
24 votes
2 answers
1k views

Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic, $$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$ This form is well-known for its connection to the symmetries of the ...
2 votes
1 answer
444 views

Automorphism group of tensor product of two graphs

Is there any relation between the automorphism group of the tensor product of two graphs $G = G_1 \times G_2$ and the automorphism groups of $G_1$ and $G_2$? I am aware of the nice results for the ...
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 ...
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 ...
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 $...
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 $$ G = \bigsqcup_{b\in B} bA.$$ ...
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_{\...
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 ...
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 ...
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 $\mu(H)=0$ ...
36 votes
1 answer
3k views

Whence “homomorphism” and “homomorphic”?

Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” (and “homomorphism” as “property of being ...
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 $\...
2 votes
1 answer
239 views

n-ary (polyadic) group "defined for tuples"

Consider an n-ary (polyadic) group, which is a generalization of the "usual" ("binary") groups to n-tuples. The definition requires appropriate standard generalizations of ...
5 votes
1 answer
2k views

Proof that the Pontryagin dual of a topological group is a topological group

I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group. It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
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 $S_n$. ...
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 $G$...
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? ...
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 ...
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 vote
1 answer
89 views

Continuous functions on HLS groupoids

I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here. Let $\Gamma$ be a discrete ...
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 ...
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 ...

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