# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

7,390
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### Inequality for word length on $\mathbb{Z}^2$

Consider the additive group $\mathbb{Z}^{2}$ and let $S\subseteq\mathbb{Z}^{n}$
be a finite set with $-x\in S$ for every $x\in S$ and $\mathbb{Z}^{2}=\left\langle S\right\rangle$. Define the ...

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### A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori

$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...

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0
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### Decompositions of groups and the existence of apartments

Let $X$ be an affine building and $G$ a group with isometric action on $X$. For any non-empty subset $\Omega$ of $X$, we denote by $P_{\Omega}$ the fixer of $\Omega$. Similarly, for any sector $\...

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### Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$

What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?

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### Grothendieck group of finite cyclic groups

Let $\mathcal{C}$ be the set of isomorphic classes of all finite cyclic groups $[C_n]$, with $C_n \cong \mathbb{Z}/n\mathbb{Z}$, $n \in \mathbb{N}_0$.
Define relations on $\mathcal{C}$ as follows: if $...

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1
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173
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### Is a Lie subgroup whose center is closed, a closed subgroup itself?

I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...

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2
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169
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### Is an extension of two abelian residually finite groups a residually finite group?

We have a short exact sequence $1 \to H \to G \to K \to 1 $, where $H$ and $K$ are abelian residually finite groups. My question is: it is true that $G$ is then a residually finite group?.

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### Centralizer of each element of a subgroup contained in the normalizer of the subgroup

Let $G \leq \operatorname{Sym}(\Omega)$ be a permutational group. A base for $G$ is a sequence of elements of $\Omega$ whose pointwise stabilizer is the identity. A base is called irredundant if no ...

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### The largest abelian subgroups of a Lie group [duplicate]

Let $G$ be a semisimple Lie group. Denote by $d(G)$ the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$, and let $c(G)$ denote the maximal integer $q$ such ...

4
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### Existence of intermediate field extensions for tamely ramified p-adic extensions

Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...

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### Solution to commutator equation in semisimple algebraic group

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and $\SL_n$ and $\GL_n$ the special and general linear groups over $K$. Let $\Phi \in \GL_n(K), H \in ...

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259
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### Using the Lehmer quintic to solve $11$-degree equations and higher?

(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...

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### Show that $\beta$ is algebraic over F($\alpha$) [closed]

Question: Let E be an field extension of F, and let $\alpha,\beta \in E$. Suppose $\alpha$ is transcendental over F but algebraic over F($\beta$).
My solution:
Let F $\subseteq $ E, a,b $\in$E and ...

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105
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### Convex subsets in abstract groups

Consider a group $G$. A norm on $G$ is a function $\|\cdot\|\colon G\to\mathbb R_+$ with (*) $\|gh\|\le\|g\|+\|h\|$ and $\|g\|=\|g^{-1}\|$ and $\|1\|=0$; the space of norms is a convex in $\mathbb R_+^...

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### The largest abelian subgroups of a Lie group

Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\...

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### Examples of groups with a positive homogeneous presentation without the Haagerup property or not of type $F_\infty$

I am looking for groups with a certain presentation that do not have the Haagerup property or are finitely presented but not of type $F_\infty$ (meaninig that for some $n\geq 3$ we cannot find any ...

55
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23
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### Results from abstract algebra which look wrong (but are true)

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...

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### Stable equivalence of generating sets of a finitely-generated group?

This question came about when I was naively considering to what extent generating sets for finitely-generated groups are unique.
Let $G$ be a group and let $\phi_1, \phi_2 : F_k \to G$ be two ...

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210
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### Comparing cohomology of a total complex with the cohomology of semidirect product

$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...

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### Characteristic subgroups of a finite abelian $2$-group

I have recently stumbled across the problem of describing the characteristic subgroups of a finite abelian group. With some discussions with some mathematicians in my lab, I managed to obtain a "...

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307
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### Geometric properties of the adjoint action of a reductive group

$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced ...

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### A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics

I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...

2
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157
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### Existence of fully supported element in a finite-dimensional vector space over $\mathbb{F}_p$ (and in finite abelian groups)

Let $V$ be an $n$-dimensional vector space over $\mathbb{F} = \mathbb{Z} / (p)$, the field of $p$ elements, $p$ a prime, with $\{v_1, \dotsc, v_n \}$ a basis for $V$. An element $x \in V$ is called &...

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1
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149
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### Finite quotients of $p$-adic congruence subgroups of $\operatorname{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime and let $ \SL^1_2(\mathbb{Z}_p)$ denote the kernel of the natrual surjective morphism $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p\mathbb{...

2
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### Characters of alternating groups

I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...

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1
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179
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### On the refined minimal ramification problem for $p$-groups

Let $p$ be a prime. The minimal ramification problem is to ask whether or not every finite $p$-group $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly ...

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### Asymptotic number of permutation representations of a given group

Let $G$ be a finitely generated group.
I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S_n$.
Equivalently this is ...

9
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1
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### Yang-Mills algebra and lower central series of surface groups

Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...

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2
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### On V. Arnold's trinities regarding PSL(2,5), PSL(2,7), and PSL(2,11)?

Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}.$ Then the following functions for levels $...

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108
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### Fractional group index?

The Hecke group of level two, $\Gamma_{0}(2)$, is an index-$2$ subgroup of the Fricke group of level two, $\Gamma_{0}^{+}(2)$, i.e. $\left[\Gamma_{0}^{+}(2):\Gamma_{0}(2)\right] = 2$. The index of $\...

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447
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### Moments of character degrees - is this result new or folklore?

Context
$\DeclareMathOperator\cp{cp}\DeclareMathOperator\AM{AM}\DeclareMathOperator\A{A}$For a finite group $G$ and $k\in\mathbb R$, define
$$
m_k(G) = \frac{1}{|G|} \sum_{\pi\in\widehat{G}} (d_\pi)^{...

3
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1
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### Reference request: Serre's Groupes discrets

I'm reading some articles and at some point they both reference:
J-P. Serre: Groupes discrets (in collaboration with H. Bass),
Collège de France, 1969
However I have trouble finding this reference. ...

5
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542
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### The relation $x \sim g x g$ in groups

While thinking about item (2) in Standard or good names for relations between maps, I thought I'd look at the relation $x \sim g x g$ in groups.
Going through all small groups of order at most 64, it ...

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### Homomorphism group of the additive group of rational numbers $\mathbb{Q}$ into quasi-cyclic group $\mathbb{Z}(p^\infty)$ [duplicate]

I read somewhere that the group of homomorphisms from the additive group of rational numbers $\mathbb{Q}$ into the quasi-cyclic group $\mathbb{Z}(p^\infty)$ is isomorphic to the additive group of ...

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### Does the visual boundary of any one-ended Cayley graph contain at least three points?

Let $\Gamma$ be a Cayley graph of a finitely generated group. We can define the visual boundary of $\Gamma$ with respect to some base vertex $b$, denoted $\partial \Gamma$, as the set of geodesic rays ...

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### Quantizing the size of a pro-$p$ group

Let $p$ be a prime number and $G$ be a pro-$p$ group (not necessarily powerful). Let $\Omega$ denote the completed group algebra $\mathbb{F}_p[[G]]:=\varprojlim_N \mathbb{F}_p[G/N]$, where $N$ ranges ...

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### Ways to prove that $n$-component Brunnian link is nontrivial

The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...

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### What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\...

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### Ramifications in Galois closures of number fields

Let $K/\mathbb{Q}$ be a finite Galois extension, and $L/K$ an infinite Galois extension such that the Galois group $ \operatorname{Gal}(L/K) $ is isomorphic to a closed subgroup of $ \operatorname{GL}...

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### Question on a theorem about group extensions

I am reading the chapter Second cohomology groups of Continuation of the Notas de Matemàtica.
Part 1 in theorem 1.2 tells us that
Let $E : 1 \to A \xrightarrow{i} X \xrightarrow{f} G \to 1$ be an ...

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### What is the least $n\ge1$ for which there is an $n$-dimensional closed flat manifold with perfect fundamental group?

In answer to the question "Is there a flat manifold with trivial first homology?" I proposed choosing a finite perfect group $P$ and a surjection $\phi:F\to P$ where $F$ is a free group of ...

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### Structure theorem for finitely generated profinite abelian groups

Is there a structure theorem for finitely generated profinite abelian group like a structure theorem of f.g. abelian group?

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### One question on linear combinations of roots of unity

For $n \geq 1$, I want to find all solutions $x_i$ of the equation
\begin{equation}
\begin{array}l
x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\
x_i^2 = 1, i=0,1,2\dotsc,n-1 \\
\...

2
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### Almost conjugate subgroups of compact simple Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...

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### In search of a quick proof that groups acting freely on $\mathbb R$-trees are linear

Many years ago I had the idea to use non-standard analysis to prove that a group acting freely on an $\mathbb R$-tree must be linear. The heuristic went like this:
A non-standard model $G^*$
of the ...

1
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0
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192
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### Theory of group representation for compact groups

I write here because some experts could help on that. It is very well known (at least for me) many reference books on linear representations of finite groups (for instance, the very classical and ...

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129
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### Cohomology of a countable directed union of groups

It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a ...

12
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2
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306
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### Conjugacy classes as left Kan extension of forgetful functor

Let $\mathbf{Set}$, $\mathbf{Grp}$, and $\mathbf{Grp}^{\rm conj}$ denote the categories of sets and functions, groups and homomorphisms, and groups and homomorphisms up to conjugation, respectively. (...

7
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448
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### What are the 4-dimensional complex representations of the real group GL(4,R)?

GL(4,R) is the group of all real nonsingular 4x4 matrices.
What are the 4-dimensional complex representations of this group?

3
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### Ways to tell from residues modulo prime factors if $z$ is below half point

Let $N=\prod_{k=0}^{k=m}{ p_k }$ be a square-free odd integer where $p_k$ is a prime. If we are given any integer $g$ such that $0<g<N$, it is very easy to tell if $g < \frac{N}{2}$ or not. ...