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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.

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43 views

Exponent of the group $GL(n,\mathbb F_2)$

Let $A\in GL_n(\mathbb F_2)$ be an element of order greater than or equal to $2^n-1$ . Then is it true that order of $A$ is $2^n-1$ ? I know that $|GL_n(\mathbb F_2)|=(2^n-1)(2^n-2^2)...(2^n-2^{n-1})$...
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19 views

Partition of a conjugacy class to conjugacy classes of a normal subgroup

Let $G$ be a group, $H$ be a normal subgroup of $G$, and $O$ a conjugacy class of $G$ contained in $H$. Consider $O = \cup_{i = 1}^{n}O_i$ the partition of O into conjugacy classes of $H$. Show that ...
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2answers
46 views

Number of normal subgroups of order $p^s$ of a $p$-group

Let $G$ be a $p$-group. Show that the number of normal subgroups of $G$ that have order $p^s$ is $1$ $mod(p)$. I think I have to use Sylow theorems and the fact that every subgroup of order $p^s$ is ...
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0answers
39 views

Computational complexity of sizes and number of orbits of a group acting on a set

Given a group action of a group $G$ on a set $X$, is there any way to relate the number of orbits, i.e. $|X/G|=|\{\{g\cdot x:g\in G\}:x\in X\}|$, to the sizes of the orbits, i.e. $|\{g\cdot x:g\in G\}|...
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2answers
37 views

Show up to equivalence that there is only one group action of $\mathbb{Z}_{3}$ on $X=\left \{ 1,2,3 \right \}$.

Show up to equivalence that there is only one group action of $\mathbb{Z}_{3}$ on $X=\left \{ 1,2,3 \right \}$. I am not sure where to begin with such a proof. I would assume it is proof by ...
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1answer
43 views

Question on group actions

Here's what I need to show: Consider the action of $G$ on itself by conjugation. In other words, if $g,a \in G$, then the action of $g$ on $a$ is defined by $g*a = gag^{-1}$. Then show that if $...
3
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1answer
19 views

Group action on Grasmannian

A book i'm reading introduces the notion of a group $G$ acting on a set $X$ and then lists these two examples. I have two questions: Example 1) The group $GL(V)$ of linear bijections from a vector ...
2
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1answer
72 views

Group action on a set that has smaller cardinality than group's one

I have a question that maybe has a trivial answer which I couldn't find though. Let's consider an action of group $\mathbf G$ on a set $\mathsf X$ (i.e. homomorphism $ \psi :\mathbf G \to S(\mathsf ...
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2answers
26 views

Find the orbits of the set $X=\left \{ 1,2,3 \right \}$ under $S_{3}$. [closed]

Find the orbits of the set $X=\left \{ 1,2,3 \right \}$ under $S_{3}$. The answer is $O_{1}=O_{2}=O_{3}=\left \{ 1,2,3 \right \}$, but I do not understand why?
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1answer
18 views

Correspondence between a group action of $G$ on a set $X$ and the homomorphism $G\to S_{X}$.

How do I show such a correspondence? I know that a group action is a map $\phi: G\times X\to X$ defined by $\phi(g,x)=gx$. Any hints to connect this to the homomorphism? Edit: Following AOrtiz's ...
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1answer
26 views

Abstract Algebra. Stabilizers

i understand parts (a) and (b) I'm trying to prove part (c) This is what i have in mind so far. am i on the right track or am i wrong? Please help.
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2answers
29 views

Sigma algebra generated by group orbits

Let $G$ be a group of bijective transformations of a set $X$. For all $x\in X$, the orbit of $x$ is the set $$G(x)=\{gx:g\in G\},$$ where $gx\in X$ is the result of the action of $g$ on $x$. A set $...
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24 views

Is $C(x)=G_{x}$ or $C(x)\leq G_{x}$?

Let $X$ be a $G$-set. In my class notes, the professor wrote the centralizer of an element $x\in X$ is equal to the stabilizer of a group $G$ acting on the set $X$. The textbook says up to subgroup ...
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1answer
51 views

What will be the orbit of the group action $G \times X \to X$ by $(g,H)=ghg^{-1}$?

What will be the orbit of the group action $G \times X \to X$ by $(g,H)=ghg^{-1}$ where $G$ is any group and $X$ is the set of all the subgroups of $G$? If $H$ is normal then the orbit of $H$ will be ...
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13 views

Stabilizer of action of $GL(N, \mathbb{R})$ on $Herm( \mathbb{C}^N)$ by changing basis

There is a group action $$GL(N, \mathbb{C}) \times Herm( \mathbb{C}^N) \rightarrow Herm( \mathbb{C}^N) $$ $$(A, X) \mapsto A^*XA $$ Metric signature of form $W$ is pair of integers $(p, q) (X)$. ...
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32 views

Stabilizer of action of $GL(N, \mathbb{R}) \times GL(N, \mathbb{R}) $ on $Mat_N (\mathbb{R})$

There is a group action $$GL(N, \mathbb{R}) \times GL(N, \mathbb{R}) \times Mat_N (\mathbb{R}) \rightarrow Mat_N (\mathbb{R})$$ $$(A, B, X) \mapsto AXB $$ I would like to find stabilizers for this ...
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59 views

Group action on $\mathbb{S}^{1}$ circle

Consider the natural group action of $O(2)$ on the set of pairs of vectors of the $\mathbb{S}^{1}$ unit circle. What is it's orbit for the group $O(2)$, give a total invariant.? Same questions of $...
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1answer
71 views

When mathematicians study specific classes of groups how do they chose which realization of them to use?

By cayleys theorem every group $G$ is isomorphic to a permutation group over the elements of $G$ yet it seems that in many instances trying to express certain groups in this way only makes it more ...
3
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1answer
64 views

Question on group action of $\mathbb{Z}/6\mathbb{Z}$ on a regular hexagon

I was working with $\mathbb{Z}/6\mathbb{Z}$ acting on a regular hexagon where $\bar{n}$ acts as rotation by $2n\pi/3,$ clockwise. By labeling the vertices (I started with $1$ at the top left corner ...
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24 views

Proof check of bijection between actions $G\to X$ and homomorphisms in $\operatorname{hom(G,S_X)}$

We can define a mapping $\Phi$ from the group actions of a group $G$ on a set $X$ to the elements of $\operatorname{hom}(G,S_X)$ as follows: let $\alpha:G\times X\to X$ be a (left) action of $G$ on $X$...
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50 views

$\text{SL}(2,\mathbb{R})$ acts on the hyperbolic space by isometries

Let $H:=\{(x_0,x_1,x_2)\in\mathbb{R}^3\mid -x_0^2+x_1^2+x_2^2=-1\}$ be the hyperbolic space with metric $g_{hip}$ induced by the Lorenz inner product $g_{Lor}=-dx_0^2+dx_1^2+dx_2^2$. Find a bijection ...
5
votes
1answer
39 views

Find group $G$ and action of $G$ on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \setminus \partial M$, open Mobius strip

I want to find a group $G$ and an action of $G$ on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \setminus \partial M$, where $M$ is the Mobious strip, and $\partial M$ is its boundary, a ...
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1answer
43 views

Exercise on categories of $G$-set

The following is exercise 6 from chapter I of Mac Lane and Moerdijk's Sheaves in Geometry and Logic: I'm currently having trouble with point $(b)$, probably because of my little ability with group ...
2
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1answer
27 views

Orbit of conjugation on subgroups of $D_8$

Let $X$ be the set of all subgroups of $D_8$ with order $2$. For fixed $g\in D_8$, and for all $x\in X$, conjugation by $g$ is defined by $$x\mapsto gxg^{-1}$$ What is the orbit of this group action? ...
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98 views

Impossible permutations of the Gear Cube

If you're familiar with the group properties of the Rubik's Cube, you will probably know that, under the action of the standard moves, all possible permutations of the (unoriented) edge pieces are ...
3
votes
1answer
52 views

Computing fixpoints of noncrossing matchings of $2n$ points under rotation.

For a definition of the cyclic sieving phenomena, see "The cyclic sieving phenomenon: a survey", by B. Sagan. A matching of $[2n] = \{1,2,\dots, 2n \}$ is a graph with vertex set $[2n]$ and with $n$ ...
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1answer
46 views

What are the orbits of the $SO(n-2)$ action on $S^n?$

Let $S^n \subset \mathbb{R}^{n+1}$ be the standard sphere. Consider the $SO(n-2)$ action on $S^n$ that fixes the first three coordinates. What is the dimension of a regular orbit, i.e, a orbit with ...
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1answer
38 views

Classification discrete subgroups of the general affine group.

Let $V$ be a real vector space of dimension $n$ and consider $GL(V)\ltimes V$ the general affine group of $V$. I would like to know about the classification of discrete subgroups of $GL(V)\ltimes V$....
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1answer
61 views

Lie algebra generated by matrices

Consider the semi-direct product $GL_2(2,\mathbb{R})\times \mathbb{R}^2$ acting on $\mathbb{R}^2\ni z$ by: $$ (A,a)\cdot z = Az+a $$ and where the Lie bracket rule is $[(A,a),(B,b)]=([A,B],Ab-Ba)$. I ...
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1answer
39 views

Show that the alternating group $A_6$ cannot act transitively on a set of 24 elements

I am trying to solve the problem in the title. My attempt is the following: Let $S$ be a group of order 24 and suppose $G = A_6$ acts transitively on $S$. Then , if we consider some $s \in S$, by the ...
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31 views

Action of $GL_n(k)$ on the space of polynomials in $n^2$ indeterminates

I'm having a hard time with a proof where the author didn't specify the action. He just wrote: "Note that $GL_n(k)$ acts on $k\big[X_{1,1},...,X_{n,n}\big]$ by substitution and leaves $k\big[X_{1,1},.....
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1answer
64 views

Discrete subgroup of topological group

I want to find textbook which contains the next proposition(?). I think it is true, but I can't find proof of that. Please teach me a textbook in which the next proposition(?) are proved. ...
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37 views

Show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is an algebraic variety

I'd like to show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is a variety. Is it even true that $\mathbb{H} = \{ x + iy : y > 0 \}$ is an algebraic variety? There's no metric, so it's ...
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0answers
18 views

Orbit space of an action over Simplicial complex

I'm studying the action of a group $G$ on a simplicial complex. I'm looking for the orbit space, or a fundamental domain for the action. There is a manner to define the orbit space like simplicial ...
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1answer
25 views

Show that $\sigma(a)\ne a,\forall\sigma\in G-\{1\}$ and all $a\in A$.where $G$ is abelian, transitive subgroup of $S_A$

Assume that $G$ is an abelian transitive subgroup of $S_A$. Show that $\sigma(a)\ne a, \forall\sigma\in G - \{1\}$ and all $a\in A$. I know what transitive subgroup is. I just can't figure out ...
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1answer
50 views

How to draw the picture of a closed horocycle in $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$?

I am trying to draw the image of the path $[0,1] + \frac{1}{7}i \in \mathbb{H}$ under the image of $\text{SL}_2(\mathbb{Z})$. This is the example of a horocycle - the image of a horocycle under the ...
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1answer
39 views

Ordering states for an integer-incrementing game.

There is a game played with a one-dimensional array of non-negative integers such as $$\underline{1}\;\underline{2}\;\underline{0}\;\underline{0}\;\underline{1}$$ Given such an array of $n$ numbers,...
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2answers
84 views

$G$ acts on $\operatorname{Hom}(G,K)$ by conjugation

I am working on an example of the Drinfeld Double of the Group Algebra and stumbled upon the book On Characters of Finite Groups. My issue is with 8.1.1, page 192 of the PDF (relevant part here). It ...
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1answer
36 views

Find all transitive $G$-sets up to isomorphism, where $G\in \{S_4,D_{10},\mathbb{Z}_{10},S_3\times \mathbb{Z}_4\}$.

I know that every transitive $G$-set $X$ is of the form $G/ H$ for some subgroup $H$ of $G$ and I know that its transitive if for some $x\in X$ the orbit $Orb_x=X$. Do I need to find all coset spaces ...
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20 views

On the dimension of a Lie group action.

Assume $G$ is a Lie group of dimension $n$ and $M\subset\mathbb{R}^m$. Consider the group action $G\times\mathbb{R}^m\to\mathbb{R}^m$. Consider now the $p$-fold product $M^p\subset\mathbb{R}^{mp}$ and ...
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0answers
21 views

group acts on the contract operator $\iota_{X_M}$

I am working on equvariant cohomology and get some problem. Give $M$ as a manifold and the action of a Lie group $G$ acts on $M$, denote the Lie algebra of $G$ by $\mathfrak{g}$. Let $\alpha$ be a ...
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22 views

Irreducible action of a group on the tangent space of a manifold

Let $M$ be a differentiable manifold. Let $G$ be a finite group of smooth maps from $M$ to $M$. Assume that the action of $G$ induced on the tangent space of $M$ at $p\in M$ is nontrivial and ...
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4answers
556 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 $$G\times X\ni(g,x)\mapsto g.x\in X$$ ...
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4answers
1k 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 ...
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1answer
34 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 ...
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0answers
27 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, ...
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0answers
37 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 ...
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0answers
26 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. ...
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0answers
17 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 ...
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1answer
70 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 ...