Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

Filter by
Sorted by
Tagged with
2 votes
0 answers
36 views

When the fundamental group of subgraph of groups embeds?

Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
tomasz's user avatar
  • 1,204
4 votes
1 answer
110 views

Examples of Noetherian integral group ring

I need to study the integral group ring of the fundamental group of a manifold. My knowledge of group and ring theory is very limited. I am looking for some examples of groups $G$ for which $\Bbb ZG$ ...
Random's user avatar
  • 927
0 votes
0 answers
65 views

Groups $P$ of order $p^5$ with $\Omega_1(P)=P$

I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...
Antoine's user avatar
  • 143
4 votes
1 answer
250 views

A pair of non-conjugate subgroups: a simple proof

$\DeclareMathOperator\SO{SO}$Set \begin{equation} \begin{aligned} \Gamma_1 &= \left\{ I_{6}, \; \gamma_1:= \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &...
emiliocba's user avatar
  • 2,321
0 votes
1 answer
158 views

Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? [closed]

I asked this question on MSE here This question was inspired by: The influence of conjugacy class sizes on the structure of finite groups. My question is as follows: Is there a way to study the ...
Naif's user avatar
  • 35
5 votes
1 answer
208 views

In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?

Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
Linus's user avatar
  • 553
0 votes
0 answers
55 views

Finite $p$-groups of maximal class whose generators have order $p$

Let $G$ be a finite $p$-group of maximal nilpotency class that is not cyclic of order $p^2$. Then $G$ is $2$-generated, say $G=\langle a,b\rangle$. Is there a classification in the case when $a^p=b^p=...
Tommaso97's user avatar
2 votes
0 answers
42 views

On the conjugation action on the first congruence subgroup of special linear group over $p$-adic fields

Let $\mathcal{O}_{L}$ be the ring of integers of a finite extension $L$ of $p$-adic number fields $\mathbb{Q}_{p}$ where $p$ is an odd prime. Let $\mathfrak{m}_{L}$ be the maximal ideal of $\mathcal{O}...
stupid boy's user avatar
2 votes
0 answers
83 views

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 ...
Mikhail Borovoi's user avatar
4 votes
1 answer
449 views

Subgroup of p-adic units

Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product ...
Nandor's user avatar
  • 289
3 votes
0 answers
158 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, ...
John Rached's user avatar
1 vote
0 answers
59 views

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 $G$ ...
tomasz's user avatar
  • 1,204
9 votes
1 answer
325 views

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, ...
Taras Banakh's user avatar
  • 40.8k
1 vote
0 answers
146 views

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. ...
utx7563yu's user avatar
1 vote
0 answers
133 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 ...
Ramin's user avatar
  • 1,362
3 votes
1 answer
223 views

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}...
Andrea Aveni's user avatar
4 votes
2 answers
214 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 ...
tomasz's user avatar
  • 1,204
2 votes
0 answers
75 views

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 $...
Antoine's user avatar
  • 143
2 votes
0 answers
226 views

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 $H^1(G,Ad(V))$ (...
user135743's user avatar
3 votes
1 answer
288 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{...
Sebastien Palcoux's user avatar
11 votes
2 answers
696 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 ...
Stefan Kohl's user avatar
  • 19.5k
3 votes
0 answers
103 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 ...
David Gao's user avatar
  • 1,262
2 votes
1 answer
156 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 ...
RKS's user avatar
  • 533
2 votes
0 answers
91 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 ...
stupid boy's user avatar
1 vote
0 answers
127 views

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 ...
medvjed's user avatar
  • 11
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 ...
jb1403's user avatar
  • 1
4 votes
1 answer
127 views

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 $\...
Another User's user avatar
-1 votes
0 answers
38 views

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 ...
Ata Berk Saraç's user avatar
4 votes
1 answer
288 views

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 \...
user524124's user avatar
7 votes
0 answers
181 views

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 ...
Alexander Chervov's user avatar
3 votes
0 answers
135 views

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 ...
A Name's user avatar
  • 31
2 votes
0 answers
84 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 ...
Bram Cohen's user avatar
6 votes
1 answer
359 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 ...
Denis T's user avatar
  • 4,416
4 votes
0 answers
186 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/...
Mikhail Evseev's user avatar
6 votes
2 answers
397 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, $...
user44312's user avatar
  • 509
2 votes
2 answers
203 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 ...
stupid_question_bot's user avatar
4 votes
0 answers
384 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 ...
Amanuel Jissa's user avatar
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 ...
Botwinnik's user avatar
9 votes
3 answers
310 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 ...
stupid_question_bot's user avatar
7 votes
0 answers
360 views

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 ...
Jeremy Rouse's user avatar
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 $\...
Bipolar Minds's user avatar
0 votes
0 answers
92 views

Isomorphism in division algebras [closed]

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 ...
Jyoti's user avatar
  • 11
0 votes
0 answers
148 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 \...
Anna Hendrix's user avatar
5 votes
2 answers
197 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 &...
stupid_question_bot's user avatar
1 vote
0 answers
156 views

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 ...
Jean-Pierre's user avatar
0 votes
1 answer
184 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 $\...
Mahdi Teymuri Garakani's user avatar
6 votes
1 answer
405 views

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| \...
David E Speyer's user avatar
4 votes
3 answers
550 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 ...
marcos's user avatar
  • 457
4 votes
0 answers
193 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 $(...
Glasby's user avatar
  • 1,961
14 votes
0 answers
472 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 ...
Ian Gershon Teixeira's user avatar

1
2 3 4 5
159