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Questions tagged [gr.group-theory]

Questions about the branch of abstract algebra that deals with groups.

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1answer
55 views

What is the defining property of reductive groups and why are they important?

Having read (skimmed more like) many surveys of the Langlands Program and similar, it seems the related ideas apply exclusively to groups that are "reductive". But nowhere, either in these surveys or ...
0
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0answers
14 views

On class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...
5
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0answers
40 views

Relative marginal subgroup

Let $w(x_1,\ldots,x_n)$ be a word in the language of group theory. If $G$ is a group, then $w(G)$ is the subgroup generated by all values of $w$ taken at elements of $G$; $w(G)$ is the verbal subgroup ...
3
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0answers
33 views

Commutator length in connected solvable Lie groups

Let $G$ be a connected solvable Lie group and let $H$ denote ist commutator subgroup. By definition, every element $g \in H$ can be written as a product of commutators and the minimal number of ...
2
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1answer
104 views

Fixed points of the automorphisms of sporadic groups

Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ...
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0answers
194 views

Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons. ...
5
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1answer
237 views

Finite-by-torsion-free abelian groups (or compact abelian groups with finitely many components)

Here's a question I should know the answer to but don't: Suppose $1\to F \to G \to G/F \to 1$ is a short exact sequence of abelian groups with $F$ finite and $G/F$ torsion-free. Must the sequence ...
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110 views

Do the ternary braid groups arise in algebraic topology?

Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$ and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
2
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0answers
109 views

Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
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0answers
75 views

Map from coproduct of products to product of coproducts

Consider obvious (imagine generators as $k \times l$ dot rectangle) epimorphism $$p: \Bbb Z^k * \dots (l \text{ times}) \dots \Bbb Z^k \to F(l)^{\times k}$$ where $F(l)$ are relatively free groups in ...
4
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1answer
89 views

non-proper parabolic isometries of hyperbolic spaces

In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\...
7
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2answers
244 views

Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$

Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?
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0answers
61 views

Monomial Characters of Quotient Groups

The following statement provides (if true) a powerful tool for inductive proofs. Can anyone confirm if it is true: Suppose $G$ is a finite group and $N$ a normal subgroup of $G$. Is it true that if $...
0
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1answer
143 views

Countable abelian group of exponent $2$

Let $\{0,1\}^{<\omega}$ denote the subgroup of $\{0,1\}^{\omega}$ where $f:\omega\to\{0,1\}$ is a member of $\{0,1\}^{<\omega}$ if there is $N\in\omega$ such that $f(n)= 0$ for all $n\in\omega$ ...
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0answers
59 views

$2$-power-torsion elements of a group

Let $G$ be a finite group, let $P$ be one of its $2$-sylow subgroups. Let $H$ be a proper subgroup of $P$, namely $H<P$ with $H\neq P$. Is it possible that $$\bigcup_{g\in G}g^{-1}Hg=\bigcup_{g\in ...
7
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0answers
83 views

Permutation groups with diameter $O(n \log n)$

I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture: Suppose that 1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
3
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0answers
103 views

Does the Hurwitz action of the braid group on rank-into-rank embeddings tend to increase the critical points?

An algebraic structure $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. Suppose that $X$ is a self-distributive algebra. Then the positive braid monoid $B_{...
12
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1answer
401 views

Is there an algorithm to decide if a word is in a finitely generated subgroup of a free group?

Let $S$ be a finite set and $F$ is the free group on that set. Is there an algorithm which takes as input a sequence of $w,w_1,\ldots,w_k\in F$ and decides whether $w\in \langle w_1,\ldots,w_k\rangle$?...
5
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2answers
396 views

Name of a group-like structure

The late Vladimir Arnold, in Arnold, V., Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world, Bull. Braz. Math. Soc. (N.S.) 34, No. 1, ...
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2answers
119 views

Conjugacy in right-angled Artin groups

I am looking for a reference containing the following result: Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the ...
7
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4answers
656 views

Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
11
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1answer
165 views

Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters

Let $G$ be a transitive permutation group of degree $d$ having a cyclic regular subgroup $K = \langle k \rangle \cong C_d$. Let $\pi(g) = |\mathrm{Fix}(g)|$ be the permutation character of $G$ and let ...
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0answers
91 views

Has the “semidirect monoid of a semiring” been considered anywhere?

Given a semiring $S$, we get a monoid $M(S)$ as follows: The underlying set of $S$ is $S^2$ The identity element is $(0,1)$ The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
6
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0answers
226 views

Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...
3
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1answer
118 views

Do the class vector and character vector of a $p$-group determine each other?

To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$: The class vector - $v_i$ is the number of conjugacy classes of order $p^i$. The character vector - $v_i$ is the number of ...
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0answers
255 views

If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle? ...
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0answers
92 views

Are double cosets of cyclic subgroups separable in a special linear group?

Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set $$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$ Is $S$ closed in the profinite topology on $\mathrm{SL}_3(\mathbb{...
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0answers
62 views

Simple group with central extension by Prüfer group and an automorphism of infinite order

Do you know of any perfect group $G$ such that 1) $Z(G)$ is a Prüfer $p$-group for some prime $p$ 2) $G/Z(G)$ is infinite simple (or a direct product of infinitely many finite simple groups) 3) ...
3
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1answer
132 views

Join prime pseudovarieties

A pseudovariety $\mathbf{V}$ of groups is join prime if for any pseudovarieties $\mathbf{V}_1, \mathbf{V}_2, \ldots,\mathbf{V}_m$, the implication $$\mathbf{V} \subseteq \mathbf{V}_1 \vee \mathbf{V}_2 ...
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0answers
65 views

Non-zero homomorphism from a module to its ground ring

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\...
1
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2answers
210 views

Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below

I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ ...
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0answers
68 views

Is the set of fixed points of hyperbolic elements from $\mathrm{PSL}_{2}(A)$ a group?

Given a subring $A$ of $\mathbb{R}$, we can consider the set $\mathrm{PSL}_{2}(A)$ of elements in $\mathrm{PSL}_{2}(\mathbb{R})$ with entries in $A$ and the determinant of associated matrix is equal ...
8
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1answer
163 views

The extension class of a finite Heisenberg group

Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the ...
14
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2answers
697 views

Units in group rings.

Let $G$ be a finite solvable group of order $n$, and let $g_1 ... g_n$ be an enumeration of its elements. Let $a_1 ... a_n$ be a sequence of integers, such that $\sum a_i$ is relatively prime to $n$. ...
7
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1answer
148 views

Finite subgroups of $PSU(3)$

I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
9
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1answer
163 views

Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions: [$C^*$-algebras and finite dimensional ...
11
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3answers
271 views

Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$? Same question, but this time $G$ is a finite group with at most $c$...
7
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1answer
361 views

Simple groups of the same order

I heard that there are no 3 nonisomorphic simple groups of the same order. Question: Is there an elementary proof of this? In case this is not the case, here a modified question: Question: Is ...
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0answers
71 views

Coxeter group action on the product of root systems

Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \...
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2answers
188 views

Is the *unreduced* Burau representation unitary?

In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...
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0answers
230 views

A new combinatorial problem for finite groups

In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
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0answers
43 views

Properties of extendable irreducible characters to a normal Sylow subgroup

Let $G$ be a finite group with a nilpotent commutator subgroup, and denote by $mcd(G)$ the set of degrees of the irreducible monomial characters. Suppose $|mcd(g)|=2$. Furthermore, we call a monomial ...
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0answers
78 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
5
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1answer
180 views

Linear representation of the free metabelian / 2-step nilpotent profinite groups on 2 generators

Let G be the free profinite group on 2 generators, $A=G/[G,[G,G]],B=G/[[G,G],[G,G]]$, then what is the structure of the groups $A$ and $B$? I heard that $A$ is isomorphic to the group of such ($3\...
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0answers
178 views

A challenging problem on disjoint cosets

Motivated by my negative solutions (see this paper published in Chinese Ann. Math. 13A(1992)) to two open problems on disjoint residue calsses posed by A. P. Huhn and L. Megyesi [Discrete Math. 41(...
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1answer
111 views

class structure constants relation

Let $C_{j,k}^l$ ,usually called class structure constants, eg Jansen and Boon and/or JQ Chen, be the number of times the class $l$ is generated from the product of classes $j,k$ and $c_j=c_{-j}$ (a ...
2
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1answer
157 views

Symmetric subgroups of simple algebraic groups over finite fields

Let $G$ be a simply connected simple algebraic group over a field $k$. Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2). Let $H=(G^\theta)^0$, the identity ...
2
votes
1answer
177 views

$2$-adic valuation on $\mathbb Q (\sqrt{-15})$

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that $-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ denotes ...
7
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0answers
153 views

Cyclic and prime factorizations of finite groups

A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$. In Cryptology factorizations of groups are known as ...
6
votes
0answers
158 views

Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...