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Questions tagged [group-actions]

Use with the (group-theory) tag. Groups describe the symmetries of an object through their actions on the object. For example, dihedral groups of order $2n$ act on regular $n$-gons, $S_n$ acts on the numbers $\{1, 2, \ldots, n\}$ and the Rubik's cube group acts on Rubik's cube.

7
votes
4answers
541 views

Is the identity axiom in the definition of group action redundant?

The definition of a group action, as given on wikipedia, is the following: Let $G$ be a group and $X$ a set. A (left) group action of $G$ on $X$ is a function $$G\times X\ni(g,x)\mapsto g.x\in X$$ ...
12
votes
2answers
93 views

What about linearity makes it so useful?

Among all areas of mathematics, linear algebra is incredibly well understood. I have heard it said that the only problems we can really solve in math are linear problems- and that much of the rest of ...
1
vote
1answer
33 views

$PGL_n$ action on an affine variety

Let $X$ be an affine variety on which $PGL_n$ acts freely. Then how to see that over the quotient variety $X / PGL_{n}$, there is a bundle of central simple algebras $M_n(k) \times^{PGL_{n}} X$. I am ...
0
votes
0answers
26 views

Examples for sequence defintion of proper group action

Sorry for the (edited) duplicate from overflow but I did not receive any answer there. For an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, ...
0
votes
0answers
28 views

Coloring the vertices of a decagon

In how many ways can you color, up to symmetry, the vertices of a regular decagon using $q$ colors? (We are talking here about the dihedral group of order $20$). So I was thinking about maybe group ...
0
votes
0answers
23 views

Transitivity of the action in definining principal bundles

In the wikipedia article https://en.wikipedia.org/wiki/Principal_bundle the definition of a principal $G-$bundle $\pi:P\rightarrow X$ demands that the action of $G$ on $P$ to be free and transitive. ...
0
votes
0answers
15 views

Identification of equivariant maps

Let $C_n$ be a cyclic group of order $n$ and $X$ be a finite $C_n$-set such that the fixed point set $X^{C_r} = X$ for the subgroup $C_r$ of $C_n.$ With this setting in hand, I want to understand the ...
-1
votes
1answer
66 views

Let $G$ be a compact Lie group acting on $M$. If $M/G$ is compact, does it follow that $M$ is compact? [closed]

Let $G$ be a Lie group which acts smoothly, freely and properly in a smooth manifold $M$. If the group $G$ is compact and the quotient smooth manifold $M/G$ is compact, is $M$ compact? • I proved ...
0
votes
1answer
37 views

Orbit space of $\partial D\times D$ under the action of $S^1$ by multiplication

Let $D$ be the unit disk, $D=\{z: |z|\leq 1\}$. Let $S^1$ act on $\partial D\times D$ by pointwise multiplication. What is the orbit space of this action? I think that since $\partial D=S^1$, this ...
2
votes
1answer
28 views

Action of $S_{d}$ in $V^{\otimes d}$

In some books is defined an action of $S_{d}$ in $V^{\otimes d}$ as $\sigma (v_{1}\otimes \cdots \otimes v_{d})=v_{\sigma^{-1}(1)}\otimes \cdots \otimes v_{\sigma^{-1}(d)}$ but then $\tau \sigma(v_{1}...
2
votes
0answers
37 views

Line bundles associated to principal circle bundles

Let $\pi: P \rightarrow B$ be a principal circle bundle over $B$ and $\rho: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ an effective left action. Then, one can associate to the bundle $\pi$ a ...
3
votes
1answer
36 views

Computing orbits of a $S_5$ group action

Consider the group action of $S_5$ on $(\mathbb{Z}/5\mathbb{Z})^6$ given by $$(12345)\colon (a,b,c,d,e,f)\mapsto(b,c,d,e,a,f)$$ $$(12)\colon (a,b,c,d,e,f)\mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$ Do we ...
2
votes
0answers
22 views

Local compactness and actions

Let $X$ be a Hausdorff topological space and $G$ a group acting continuously on $X$. We denote the group action as $(g,x) \rightarrow gx$. Let $x\in X$ and $U$ a neighborhood of $x$ such that the set $...
1
vote
1answer
31 views

Mapping Class Group acts properly discontinuous; Alexander method

Let $S$ be a closed surface of genus $g$. The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" (p.59) (roughly) states that if $c_1,c_2$ are two filling curves in minimal ...
2
votes
0answers
21 views

Minimality of group action on compact topological space implies that finite orbit of any non-empty open set covers the space

I have that X is a compact topological space and G is a group that acts on X minimally in the sense that $\forall$ x $\in$X, $\overline{\cup_{g\in G} (g.x)}$ = X. I want to show that given any non-...
1
vote
1answer
36 views

Stabilizer and Set of Fixed Points for a Given Action

I've been reviewing some group theory; particularly group actions. I've been working on the following problem. Let $F(\mathbb{R})=\{f:\mathbb{R}\to\mathbb{R}\}$ denote the collection of all real-...
1
vote
0answers
25 views

The dual Lie algebra in the context of Hamiltonian action

Let $t$ be a Lie algebra. what is the precise structure of the dual lie algebra $t^*$ when we are considering the momentum map associated to a Lie group action on a symplectic ...
3
votes
0answers
52 views

Classifying the orbits of the natural $\text{GL}(V)$-action on the exterior power $\bigwedge^k V$

Let $V$ be a real $d$-dimensional vector space, and let $1 < k < d$. Consider the following action of $\text{GL}(V)$ on $\bigwedge^k V$: $(T,\omega) \to (\bigwedge^k T) \omega$. Can we ...
2
votes
0answers
69 views

Determining the isomorphism classes of these symmetry groups (exterior algebra)

Let $V$ be a real $d$-dimensional vector space. Let $\omega \in \bigwedge^kV$ be a fixed non-zero multivector for some $1 < k < d$. Define $ G_{\omega}=\{ T \in \text{Aut}(V) \, | \, (\bigwedge^...
2
votes
0answers
31 views

free $G$-action on a topological space

Does existence of a free finite $G$-action on a topological space invariant under homotopy? In other words, let $X$ be a free $G$ topological space which is homotopic with a topological space $Y$. ...
3
votes
0answers
45 views

How to think of these different $K$-points of a scheme and Galois action

The eventual goal of this question is to better under the action of a Galois group on a scheme defined over a field $K$. Right now I just want to understand how to think about $K$-points and their ...
1
vote
1answer
27 views

Kernel of group action on set of cosets by left side multiplication

I was reading this article that explains some ways you can show a certain group is not simple. One of the methods described is by observing the action of the group $G$ on the set of cosets $G/H$ (for ...
2
votes
0answers
46 views

Burnside groups : a few questions

Let $G=B(n,e)=F_n/\langle\langle F_n^e\rangle\rangle$ be the Burnside group on $n$ generators with exponent $e$, i.e. the quotient of the free group on $n$ generators $F_n$ by the normal subgroup ...
0
votes
1answer
28 views

Lie group actions

Let $G$ be a group. 1- What is the kerner of (non-faithful) transitive group action (since the stabilizer is a subgroup of G and the kernel is normal subgroup of G and there exist only one stabilizer ...
2
votes
1answer
44 views

Let$G$ be a group of order $12$ with $4$ Sylow $3$-subgroups, affording permutation representation $\phi : G \to S_4$. Prove that $ker \phi = 1$.

Let$G$ be a group of order $12$ with $4$ Sylow $3$-subgroups, affording permutation representation $\phi:G \to S_4$. Prove that $ker \phi = 1$ This is what I am trying to understand from Dummit and ...
0
votes
1answer
38 views

Given groups $H, K$ and $\cdot : H × K \to K$ a group action, is $a \cdot b × a \cdot c = a \cdot (b × c)$?

Given groups $H, K$ and $\cdot : H × K \to K$ a group action, is $a \cdot b × a \cdot c = a \cdot (b × c)$? Here a group action is $\cdot : H × K \to K$ such that for all suitable elements $h \cdot (...
11
votes
1answer
322 views

Does the following object have such property?

Let $S_n$ denote the set of bijections on the set $M = \{1, 2, ... , n\}$. Suppose that a set $\Omega \subset S_n$ satisfies the following condition: there is $k \leq n$ such that, for each nonempty ...
0
votes
1answer
33 views

Different definitions of proper actions

I want to compare three definitions for an action $G\times X\to X$ to be proper: For each $x\in X$, there is a neighborhood $U\subseteq X$ such that $gU\cap U=\emptyset$ for all but finitely many $g\...
1
vote
1answer
99 views

Moment map for Hamiltonian action $\Bbb S^1 \circlearrowright \Bbb C^n$.

Consider $\Bbb C^n$ with the usual $\newcommand{\d}{{\rm d}}$symplectic form $$\omega =\frac{i}{2}\sum_{k=1}^n \d z^k \wedge \d \overline{z}^k$$and the action $\Bbb S^1 \circlearrowright \Bbb C^n$ ...
4
votes
1answer
67 views

Prove that $H\cap gHg^{-1}$ is normal

This problem is from a Ph.D Qualifying Exam from algebra: Let $G$ be a finite group and let $H \le G$ with index $[G:H]=n$. If $H$ is a maximal subgroup of $G$ and $H$ is abelian, show that $H\cap H^...
2
votes
3answers
33 views

Orbits of $S$ constitute a partition of $S$.

Let $G$ be a group of permutations of a set $S$. Prove that the orbits of the members of $S$ constitute a partition of $S$. Things I know: Intuitive definition of an orbit: The orbit of $x$ is "...
9
votes
0answers
87 views

Does the space of matrices above rank $k$ admit a transitive Lie group action?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector space ($d \ge 4$). Let $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \text{End}(...
1
vote
2answers
36 views

Induced action is proper discontinuous

Let $f:X\to Y$ be a surjective map, and let $G$ act on $X$ such that for each $g\in G$ and $x,x'\in X$ $f(x')=f(x)$ implies $f(g\cdot x)=f(g\cdot x')$. Further assume that the group action on $X$ is ...
2
votes
1answer
59 views

Can a group action be defined by its orbits?

I am trying to solve the following qual study problem: Let $p$ and $q$ be distinct primes, and let $G$ be any group of order $pq$. Show that for every integer $n \geq pq$, there exists a set $\...
0
votes
1answer
29 views

How can I show the action of left translation by a topological group $G$ on itself is a continuous action using the ordinary definition of continuity?

Let $G$ be a topological group and let the action $f: G\times G \to G$ be defined by $f(g,g')=gg'$. I want to show that this action by left translation is a continuous action by showing that if $U \...
2
votes
1answer
25 views

(Soft question) Classify cohomogeneity one actions up to what?

This is a question about cohomogeneity one actions of a compact Lie group $G$ on a Riemannian manifold $M$, such that $G$ acts via isometries. Many articles have been published about classifications ...
1
vote
0answers
34 views

Normalizer, stabilizer and orbits

I know relation between normalizer of a subgroup $H$ of a group $G$ and the stabilizer of one point $x$ of a set $Ω$ on which the group acts. Given that$$ \gcd([G:\operatorname{Normalizer}(H)], [G:\...
0
votes
1answer
26 views

$SU(2)$ acts transitively on $\mathbb{CP}^1$

How can one proof rigorously that the action of $SU(2)$ on $\mathbb{CP}^1$, where $\mathbb{CP}^1$ is the complex projective space, is transitive? i.e., that for any $u, v \in \mathbb{CP}^1$, there ...
15
votes
1answer
198 views

Grothendieck's Galois theory: fundamental theorem

I've been learning about Grothendieck's Galois theory, and I just haven't been able to understand the fundamental theorem properly. Let's phrase the fundamental theorem in the case of fields: Let $...
0
votes
0answers
16 views

Infinitesimal left actions are Lie algebra ANTIhomomorphisms?

Before I ask the question I clear up some notation. Throughout this question $M$ will be a smooth $n$ dimensional manifold, $G$ a $k$ dimensional Lie group and $G\times M\rightarrow M$ a smooth left ...
3
votes
1answer
30 views

Fixed points of finite order mapping classes

I am currently working myself through the proof that every finite order element of the mapping class group $\mathrm{MCG}(S_g)=\mathrm{Hom}^+(S_g) / \mathrm{Hom}_0(S_g)$ of a closed, hyperbolic, genus $...
0
votes
1answer
19 views

Relation between $G$-orbits and Cycle Decomposition of a Permutaion.

Let $X_n=\{1,2,...,n\}$, $\delta \in S_n$. Write $G=(\delta)$ and assume $G$ acts on $X_n$. What is the relation between $G$-orbits of $X_n$ and cycle decomposition of $\delta$?
0
votes
0answers
42 views

Proving that the group action of $\mathbb{Z_p}$ over a set $X$ is well defined

I am trying to prove Cauchy theorem for groups which states that: If $G$ is a finite group, $|G| =n$, and $p$ a prime number which is a divisor of $n$, then there exists $g\in G$ such that $g^p =1$. ...
1
vote
0answers
19 views

Freeness of an action of $G$ on $X\leftrightarrow$ freeness of some $\mathbb Z[G]$-module?

Consider a group $G$ acting on a set $X$. The action of $G$ is free if and only if $g(x)=x\Rightarrow g=e$. Consider the $\mathbb Z[G]$-module $M(X)$ given by the free $\mathbb Z$-module generated by $...
0
votes
1answer
26 views

How to arrive at this quotient topology on an orbit space?

My question relates to group actions on manifolds. If $G$ is a group and $M$ is a manifold, then the set of orbits $M/G$ is equipped with the quotient topology. I consider the following example: The ...
1
vote
1answer
39 views

$G$-set isomorphisms for free action on some product

Let $G$ be a finite group, $X$ a finite set on which $G$ acts freely. Consider the set $G\times X$ with diagonal action of $G$, i.e. $ h.(g,x) = (hg, h.x).$ This is also free. For the orbits, we have $...
0
votes
0answers
18 views

isotropy group of a point

if we have a mapping on C2 defined by (z1,z2) ---> (t^m1 . z1 , t^m2 . z2) where t belongs to S1 , ma,m2 belong to Z. first i need to prove that this mapping is an action of S1 on S3 belong C2 for ...
2
votes
0answers
28 views

Isometric action of $SL_{2}(\mathbb{R})$ on a metric space and fixed point

I need to prove the following statement : Assume that $\mathrm{SL}_{2}(\mathbb{R})$ acts on $(X,d)$ by isometries. Assume that $x\in X$ and $a_{n}\in A=\{\begin{pmatrix} e^{t/2}&0 \\ 0&e^{-t/...
0
votes
1answer
75 views

Cohomology classes of regular level sets of moment maps

Let $(M,\omega)$ be a closed symplectic manifold. Assume $M$ is equipped with a Hamiltonian $T^r$-action, $r \in \mathbb{N}$ and let $\mu:M \to (\mathbb{R}^r)^*$ be its moment map. Let $A = \mu^{-1}(y)...
6
votes
4answers
109 views

Orbit of a matrix can generates a basis?

Let A be a matrix $n\times n$, given a n-vector $v$, what conditions over $v$ and $A$ are necessaries for $[v, Av,..., A^{n-1}v]$ will be linearly independent? For example if $v$ is a eigenvector or $...