Questions tagged [group-actions]

Question 1

Background: Definition: We say the group $G$ acts on a set $X$ if there is a homomorphism $\sigma:G\to S_X.$ Thus $\sigma(G)$ is a subgroup of $S_X,$ the group of all permutations of $X.$ ...

Question 2

Let $G$ be a complex algebraic group and let $F$ be a finite group acting on $G$ by inner automorphisms. It is easy to prove that, if $G$ is connected, then the induced $F$-action on the (complex) ...

Question 3

Let $X$ be a topological space and $G \subset \text{Aut}(X)_{\text{top}}$ a finite group acting faithfully on $X$ "nicely enough", where "nicely enough" = that we can form quotient ...

Question 4

I am working on a project that combines ingredients from geometric group theory and spectral graph theory, and I would appreciate references (papers, authors, or survey articles) that study anything ...

Question 5

Let $G$ be a Lie group acting properly and isometrically on a Riemannian manifold $(M,\mathtt g)$. By the slice theorem, we know that the tubular neighborhood $U=\exp(\nu^\epsilon G(x))$ of an orbit $...

Question 6

Let G be a group acting on a set A. Let $[x]$ denote the orbit of any $x\in A$. Also let $G_x$ denote the stabilizer of $x$. From the orbit-stabilizer theorem, the orbit of any $x\in A$ has the same ...

Question 7

I want to prove the following about the action of $SL(2, \mathbb R)$ on $\mathcal{H}^n$ by Möbius transformations, $SL(2, \mathbb R)$ acts transitively on the space of directions at $ i \in \mathcal{...

Question 8

I was reading the following 2 questions here: 1- Show that the real axis and the upper half plane are mapped to themselves under $SL(2, \Bbb R)$ 2- Prove that $SL(2,\mathbb{R})$ acts transitively on ...

Question 9

Let $M$ be a $G$-module for some group $G$, and let $H$ be a finite index subgroup of $G$. Let $M_G$ denote the coinvariants of $G$ acting on $M$. So $M_G = M / K$ where $K < M$ is the subspace ...

Question 10

Let $X$ be a topological/smooth/complex etc. manifold and $G \subset \text{Aut}(X)_{\text{top, smooth, complex}}$ a finite group acting faithfully on $X$ as homeomorphsms/diffeomorphisms/holomorphic ...

Question 11

Let $A, B$ be finite groups. Suppose $A \triangleleft B$ with $B$ transitively acting upon $A \setminus \{1\}$ by conjugation. (This implies $A \cong (\mathbb{F}_p^n, +)$.) Must there exist $C$ with $...

Question 12

Exercise 10.33 in Rotman's "An Introduction to Algebraic Topology" asks us to prove that Let $G$ be a group. If there exists a tree $T$ on which $G$ acts properly (discontinuously, in the ...

Question 13

Let $G$ be a finite group and let $M$ be a G-module. So we are given an action $$G\times M\to M$$ by $(g,m)\mapsto g.m$. Let $H^2(G,M)$ be the second cohomolgy group. Define the following action of $G$...

Question 14

I was studying about group actions. And, I read some parts of the Wikipedia article about the same. Let $G$ be a group acting on a set $X$ by action $\alpha$. We call the action free if no element ...

Question 15

Question: Does anyone know of a name for the "Mystery" property below? (Please provide a reference if available.) Request: Does anyone know of references that in some way use or mention any ...

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Questions tagged [group-actions]

Quick questions about left/right group actions [closed]

Finite group acting on a connected algebraic group

Correspondence Local Systems & $\Bbb Z \pi_1(A)$-modules and its Compatibility

Looking for authors/papers in GGT involving piecewise isometries, free group actions, and twisted Ihara zeta functions

G-invariant metrics on a Tubular neighborhood

The consequences of the orbit-stabilizer theorem

Space of directions meaning in the proof? Does it means the whole $\mathbb R$?

Acts transitively vs sending the upper half plane points to itself

Coinvariants and finite index subgroups

Comparison Invariant Cohomology with Cohomology of Quotient Manifold

Question about transitive conjugation actions

Properly Discontinuous Action of a Group on a Tree

Action on the second group of cohomology

Cayley's Theorem [duplicate]

Characterization of group actions $G \times X \to X$ by the partially applied functions $G \to X$?

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