Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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What categorical property of monoidal categories picks out the ones with duals?
Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...
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Rokhlin lemma for arbitrary infinite groups.
Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.
It is well known that if $G$ is a finite group then this action ...
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Fundamental group of punctured simply connected subset of $\mathbb{R}^2$
(This question is originally from Math.SE where it was suggested that I ask the question here)
Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning ...
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Finitely generated groups with Hölder-exotic space of ends?
The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...
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An equivalence relation on group actions
Suppose a group $G$ acts faithfully on a set $X$, or equivalently, $G$ is a subgroup of ${\rm Sym}(X)$.
By functoriality, $G$ acts on $P(X), P(P(X)), P(P(P(X))),$ etc. ($P(\cdot)$ means powerset.)
...
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Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
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Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
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(Dis)similarity between groups and Lie algebras
There are many questions which sound similar or the same for groups and Lie algebras. Some (very few, actually) of those questions have identical solutions and answers. Some have identical answers but ...
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Grothendieck's question on the Brauer group for groups
Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear ...
- 5,062
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Lower bounds on the number of elements in Sylow subgroups
I posted this question on Math.SE (link), but it didn't get any answers so I'm going to ask here. This is an edited version of the question.
Let $p$ be a prime and $n \geq 1$ some integer. ...
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What is the strongest nerve lemma?
The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology:
If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
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$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?
During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you :
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, ...
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Definition of Pin groups?
When looking into the definition of a Pin group, it turns out that there are - at least - three different ones in the literature, and they do not agree --- but thankfully all yield the same Spin ...
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How boundedly generated is $SL_3(\mathbb{Z})$?
The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
- 11.3k
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Linear groups which don't contain products of free groups
Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
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Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?
Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later ...
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Quotients of residually finite groups by amenable normal subgroups
My questions are:
Is there any group, which cannot be written as the quotient of a residually finite group by an amenable normal subgroup? Is it possible for large classes of groups?
and
Is ...
- 25.5k
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Why semigroups could be important?
There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
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〈x,y : x^p = y^p = (xy)^p = 1〉
Let $p$ be an odd prime and $G := \langle x,y : x^p = y^p = (xy)^p = 1 \rangle$. I want to show that $G$ is infinite and wonder if there is a good way to prove this. I'm familiar with the basics of ...
- 63.1k
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General bound for the number of subgroups of a finite group
I am interested in the following:
Let $G$ be a finite group of order $n$. Is there an explicit function $f$ such that
$|s(G)| \leq f(n)$ for all $G$ and for all natural numbers $n$, where $s(G)$ ...
user13040
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Does the map $\mathrm{SL}(n,\mathbb{Z}/p^2) \rightarrow \mathrm{SL}(n,\mathbb{Z}/p)$ split?
$\DeclareMathOperator\SL{SL}$ $\DeclareMathOperator\GL{GL}$The question is the one in the title: for a prime $p$, does the obvious surjection $\pi\colon \SL(n,\mathbb{Z}/p^2) \rightarrow \SL(n,\mathbb{...
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Examples of locally hyperbolic groups
It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
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Cayley graphs of $A_n.$
Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an ...
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Reference for this theorem in representation theory?
Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/...
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Generating the symplectic group
The too naive and vague version of my question is the following: given a collection of integer symplectic matrices all of the same size (say 2n by 2n), how can I tell if they generate the full ...
user5117
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When is $G$ isomorphic to $G \times G$?
Is there a finitely generated nontrivial group $G$ such that $G \cong G \times G$?
Here are some properties which such a group $G$ has to satisfy:
$G$ is not abelian (otherwise $G$ is a noetherian $\...
- 63.1k
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A result of Schützenberger on commutators and powers in free groups
It is an old result of Schützenberger that in a free group, a basic commutator cannot be a proper power. A look at the original reference
M.-P. Schützenberger, Sur l'équation $a^{2+n} = b^{2+m}c^{2+p}...
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Are unitarily equivalent permutation matrices permutation similar?
Two matrices $A, B \in \mathbb R^{n \times n}$ are called unitarily equivalent if there exists an unitary matrix $U \in \mathbb C^{n \times n}$ such that $A = U B U^{\ast}$. If in addition $U$ is a ...
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No injective groups with more than one element?
There are several claims in the literature that there are no injective groups (with more than one element), but I have not found a proof. For example, Mac Lane claims in his Duality from groups paper ...
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Where should I search for computations of group cohomology rings of not-too-complicated finite groups?
A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low
degrees, and I'd like to determine where to search for preexisting computations.
...
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When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?
A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...
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A synopsis of Adyan’s solution to the general Burnside problem?
Where can I find a high-level overview
of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent?
Additionally:
If possible, would an expert please ...
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one-parameter subgroup and geodesics on Lie group
Hi,
Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...
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In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?
Let $H$,$K$ be closed connected subgroups of a compact Lie group $G$. Let $L:=\langle H,K \rangle$ be the subgroup they generate, ie, the smallest subgroup of $G$ containing them both. Must $L$ be ...
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Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 128k
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A Shelah group in ZFC?
In his famous paper "On a problem of Kurosh, Jonsson groups, and applications" of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6640}$ for any uncountable ...
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God's number for the $n \times n \times n$-cube
This is a question about Rubik's Cube and generalizations of this puzzle, such as Rubik's Revenge, Professor's cube or in general the $n \times n \times n$ cube.
Let $g(n)$ be the smallest number $m$, ...
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Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
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A finite 2-group containing the dihedral group of order 16?
The dihedral group $D_{16}$ of order 16 has a presentation $D_{16}= \langle a,t \ | \ a^2=t^8=atat=e\rangle$.
Question: Does there exist a finite 2-group $G$ containing $D_{16}$ as a subgroup, and ...
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Does this subgroup of "even braids" have a name?
The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation ...
- 35.9k
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Connes' embedding conjecture for uncountable groups
In this topic, I will use the word uncountable group referring to groups whose cardinality is $\leq|\mathbb R|$.
Notation: $R$ is the hyperfinite $II_1$-factor, $\omega$ is a free ultrafilter on the ...
- 6,034
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Probability that a word in the free group becomes (much) shorter?
Let $w$ be a word of length $2\ell$ chosen at random on the alphabet $\{x_1,x_1^{-1},x_2,x_2^{-1},\dotsc,x_k,x_k^{-1}\}$. By the reduction $\rho(w)$ I mean what you obtain by deleting substrings of ...
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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 ...
- 25.8k
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Existence of a quasi-isometric residually finite group?
It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...
- 733
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Infinite groups with oligomorphic conjugation action
The action of a group $G$ on a set $X$ is called oligomorphic if the diagonal action on $X^n$ has finitely many orbits for each $n$.
Question: Is there an infinite (maybe even finitely generated) ...
- 25.5k
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Why are model theorists so fond of definable groups?
My PhD was on so called "pure" model theory, and my advisor was not very much interested in applications of model theory to algebra. Now I feel the need to fill in the gap, and I'd like to educate ...
- 1,031
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Number of solutions to equations in finite groups
Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.
Is it always true that the number ...
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Finitely generated Galois groups
It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number theory....
- 1,418
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How to find more (finite almost simple) groups with a given Sylow subgroup
I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle ...
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Relationship between Smith's special homology groups and equivariant homology theory
EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is ...
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