Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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2 votes
1 answer
85 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 ...
1 vote
1 answer
361 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
113 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 $\Bbb ZG$ ...
5 votes
1 answer
212 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 $B/A$. ...
0 votes
0 answers
66 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 ...
4 votes
1 answer
251 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\\ &...
4 votes
1 answer
182 views

Existence of disintegrations for improper priors on locally-compact groups

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
0 votes
1 answer
159 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 ...
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=...
2 votes
1 answer
200 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 ...
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, ...
4 votes
1 answer
451 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 ...
2 votes
0 answers
43 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}...
3 votes
1 answer
225 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}...
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 ...
9 votes
1 answer
327 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, ...
15 votes
0 answers
746 views

Are all these groups CAT(0) groups?

Given a geodesic metric space $X$ together with a choice of midpoints $m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$). Assume furthermore, that the following nonpositive curvature ...
7 votes
2 answers
3k views

On the cohomology ring of the Grassmannian

The basis of Schubert classes for the cohomology ring $H^*(\text{Gr}(m,N))$ of the Grassmannian of $m$-dimensional subspaces of $\mathbb{C}^N$ is indexed by $L(m,N-m)$, the poset of all partitions ...
1 vote
0 answers
61 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 $G$ ...
5 votes
4 answers
3k views

Number of non-Abelian groups of order $2^n$

Related to A000679 (Number of groups of order $2^n$), how many non-Abelian groups of order $2^n$ are there?
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. ...
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 ...
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 ...
3 votes
1 answer
289 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
698 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 ...
12 votes
1 answer
373 views

Does every topological group embed as a closed subgroup in an amenable group?

It is a standard result that closed subgroups of locally compact amenable groups are themselves amenable, so for example $F_2$, the free group on two generators, cannot be embedded as a closed ...
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 $...
2 votes
1 answer
294 views

$\omega$-Commuting matrices vs Stone-von Neumann Theorem

Let me first recall the Stone-von Neumann theorem that if two one-parameter groups of unitary operators $U_t$ and $V_s$ over a Hilbert space satisfy $U_tV_s=e^{ist}V_sU_t$ for every $s,t\in{\mathbb R}$...
2 votes
0 answers
227 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 $H^1(G,Ad(V))$ (...
11 votes
2 answers
2k views

Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion: Topology on the Unitary Dual Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let $...
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 ...
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 ...
2 votes
0 answers
92 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 ...
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 $\...
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 ...
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 ...
17 votes
2 answers
825 views

The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

Throughout $G$ is a finite, non-abelian group. $\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$ Let $\Irr(G)$ be the set of ...
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 \...
-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 ...
6 votes
1 answer
360 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 ...
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 ...
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 ...
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 ...
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 ...
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/...
6 votes
2 answers
398 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
204 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 ...
7 votes
1 answer
586 views

Reversible varieties

We say that a variety $V$ is reversible if for each $n>0$ and $n$-ary fundamental operation $f$, there is some $m\geq n$ and $r$ along with terms $T_{2},\dotsc,T_{r},S_{1},\dotsc,S_{m}$ such that $...
11 votes
2 answers
685 views

Where does the term "torsor" come from?

Is there a heuristic reason why principal homogeneous spaces of a group (object) $G$ (in some categories) are called $G$-torsors? Does it have anything to do with the idea of "torsion", ...
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 ...

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