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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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Corresponding space to a irreducible part of a given representation of $S_n$

My setting is the following: I have an action of $S_n$ on some $\mathbb{C}$-vector space by permuting a special basis. There I have the subspace such that every element of $S_n$ acts by its sign, ...
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what is the definition of G compact set when the space is equipped with a proper continuous group action?

I am reading a book and it doesn't gives a definition, I searched it on the internet, but no result was found.
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“Neutral element” of a group action in the set [duplicate]

I have difficulties solving the following problem, I would be grateful for any tipps or advice: a) Let $p$ be a prime and $G$ a group with $|G|=p^n$. $G$ operates on an finite set $X$ with $p \nmid |...
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1answer
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How many different ways can $\mathbb{Z}_3$ act on the set $\{1, 2, 3, 4\}$

How many different ways can $\mathbb{Z}_3$ act on the set $\{1, 2, 3, 4\}$ This is my attempted proof. Proof: Any action of $\mathbb{Z}_3$ on the set $\{1, 2, 3, 4\}$ is equivalent to a homomorphism ...
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45 views

Invariant measure of a continuous group action

Denote by $\mathcal{M}(X)$ the set of Borel probability measures of a space $X$. This set is equipped with weak$^*$ topology defined by the convergence $\mu_n\to \mu$ if and only if $\int \phi d\mu_n\...
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1answer
27 views

Does the number of actions a group can have on the left cosets of a subgroup of index $n$ tell us anything about the number of such subgroups?

I have a question asking me to show that the number of index $n$ subgroups of a group $G$ of rank $r$ is bounded above, and it offers the hint that I should consider the number of actions $G$ can have ...
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Continuous group action on a finitely generated group and compact Hausdorff space

Let $\varphi:G\times X\to X$ be a continuous action such that $G$ be a finitely generated group, $X$ be a compact Hausdorff space, and $\mathcal{U}=\{U_i\}_{i=1}^m$ be a finite open cover of $X$. ...
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Left action of a group on permutation representation

I am studying my course on permutation representation, and I am stuck at understanding the left action of a finite group on the permutation representation $F(X,\Bbb C)$. In my course it is given for $(...
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1answer
38 views

Conjugacy classes in space of trace zero 2*2 matrices

I'm trying to find the orbits when $SL_2$ operates by conjugation on $\mathfrak{sl}_2=Lie(SL_2)=\{A|\operatorname{tr} A=0\}$. I have tried to write $X\in sl_2$ and corresponding $AXA^{-1}$ for random ...
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Group action in Polynomial invariant

The following is just a basic definition in Invariant Theory, which I copied from wikipedia "Let $G$ be a group, and ${\displaystyle V}$ a finite-dimensional vector space over a field ${\displaystyle ...
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Finding generators of a group from its action on a topological space

Summary I believe I've written a geometric group theory flavoured proof with a mistake in it, but I'm struggling to see why it might be wrong. I haven't found a counter example, but it also feels too ...
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great circle and latitude preserving maps of the sphere

Let $S^2 = \{ (x,y,z)\in\mathbb{R}^3 : x^2 + y^2 + z^2 =1\}$. I would like to find all maps of the sphere sending great circles to great circles and latitudes perpendicular to the $z$-axis to ...
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On the proof of localization in symplectic geometry

I was working on the proof of Duistermaat-Heckman theorem in Introduction to Symplectic Topology by Dusa McDuff. He used a lemma called localization. It can be found on page 194. You can find the ...
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1answer
28 views

Notations for Orbits, Stabilizers, Centralizers, Conjugation Classes, and the Center of a group

I'm trying to understand these different aspects of group actions and the thing that's throwing me off the most is the notation for these sets. Additionally, when browsing other stack exchange ...
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1answer
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Existence of $S^1$-action on a vector bundle and computing its characteristic classes

The existence of an $S^1$ action sometimes helps us in computing topological invariants. For example we can compute the Euler characteristic looking at the fixed point set (see Euler characteristic ...
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1answer
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Why is set required to be finite in the Orbit-Stabilizer Theorem

In the Orbit Stabilizer Theoren (found here) we have a group acting on a finite set $X$. I don't see in the proof given how the finiteness of the set is used. I get that if one wants to write $\...
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Group $ GL(V) $ acts naturally on $ \mathscr{F} $, how to get its orbits?

Could you please explain to me the meaning of the marking? Thanks in advance.
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2answers
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How to construct a two-sheeted cover of a non-orientable surface?

Let $S$ be a non-orientable surface. Then there exists a two-sheeted covering map $p:S'\to S$ with $S'$ an orientable surface. I want to know how to construct $p$. I know that $\mathbb{R}P^2$ is 2-...
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1answer
27 views

Smooth point of the quotient space obtained by Lie group action

Let $G$ be a Lie group acting smoothly and effectively on a smooth manifold $M$ and $\pi: M \to M/G$ the quotient map. Can we find a point $p \in M$ such that an open neighborhood of $\pi(p)$ is ...
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1answer
55 views

A quotient space by group action is manifold $???$

The aim of this question is to construct a exmaple of quotient manifold. First, I set notations and difnitions. Let $\mathbb{R} , \mathbb{C}$ be real and complex numbers with usual topology. We ...
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Rewording of a question regarding actions.

Can someone re-word the following question in to more basic terms: 'Let $X$ be a finite set. A powerset $P(X)$ of $X$ is the set of all subsets of $X$. Define $\lambda$ as follows $\forall g \in Sym(...
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1answer
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Using Pólya counting to find number of conjugacy classes of $S_3$

So I know that $S_3$ has three conjugacy classes. However, I was reading about Pólya counting today, and am wondering how Pólya counting could be used to derive the number of conjugacy classes for $...
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1answer
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Is it possible to “mod” the action of a symmetric group on a symmetric operad?

I am relatively new to category theory, so only have a rough understanding of the technicalities behind operads. My understanding is that symmetric operads are defined so that they are "nicely" acted ...
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What does it mean for a group action to be linear?

Let $G$ be a group and $X$ a set. The group $G$ acts on the set $X$ if there is a map $$G \times X \rightarrow X, (g,x) \rightarrow gx$$ such that $ex= x$, $h(gx) = (hg)(x)$ for all $x \in X$ and all $...
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Irreducible actions of $\mathbb{Z}^d$ on $\mathbb{T}^n$

I am trying to understand the construction done in this reference http://www.personal.psu.edu/sxk37/pub/KKS-old.pdf by Katok, Katok and Schmidt. In the section 3.3 (p11-13), the idea is to ...
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2answers
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Burnside convolution

Let $G$ be a group. Say that an orbit is a nonempty transitive $G$-set. Let $\Xi$ be a set of finite orbits such that each finite orbit is isomorphic to exactly one element of $\Xi$. If $X,Y,Z\in\Xi$,...
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What is the action of a primitive permutation group of type SD?

My goal is to show that $Alt(5) \wr Sym(3)$ is a primitive permutation group of type SD. Let $G$ be a primitive permutation group, $N$ a minimal normal subgroup, and $H$ a point stabilizer (so $NH =...
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Algorithm to find representatives of orbits when group of permutations acts on set of mappings

Let $N$ be $\{1,\ldots,n\}$, $R=\{1,\ldots,r\}$. Consider mappings from $N$ to $R$. It is clear that there are $r^n$ such mappings. Let $G_n$ be the group generated by $n$ length cycle permutation $\...
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Dixon&Mortimer exercise involving size of orbits under subgroup of point stabilizer of given primitive permutation group

I come forth once again with a Dixon&Mortimer exercise (1.5.22), formulated as follows: let $G$ act faithfully and primitively (transitivity implicitly assumed) on finite set $A$ with $|A| \...
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continuous action on the sphere

I am wondering if the following action defines a continuous action on (a quotient) of $S^2$. Consider the rotation action of $S^1$ on $S^2$ along the vertical axis containing the north and south pole,...
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Confused to prove Group Actions

I want to ask about Group Action. This is the definition of group actions. Let $G$ be group and $\Omega$ is a set. Group action $G$ on $\Omega$, defined by if there exist mapping \begin{eqnarray} \...
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1answer
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Is there a tensor product of $G$-sets?

We can take the tensor product of two vector spaces, and the tensor product of two modules. I'm wondering if the same can be done for group actions. Let $G$ be a group which acts on two sets $X$ and ...
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1answer
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What are the orbits for this action of $\mathrm{SO}(2)$ on $S_3$?

The group action defined by: $\Phi: SO(2) \times S_3 \rightarrow S_3$ such that: ($A, r$) $\rightarrow$ [$ \mathbb{1} \otimes A$]$\cdot$[$r$], where A is a standard matrix $2 \times 2$ of 2D rotations,...
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1answer
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Why minimal actions of a Abelian group is not proximal unless the action be tivial?

Let $G$ be a discrete abelian group which acts minimally on a compact (for comfort, compact and metrizable) space $X$ by homeomorphisms. If the action is proximal in addition then $X$ must be a ...
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Compute the size of the orbit under a finite group action

When $n = 3$, consider the three-dimensional tensors $X, X' \in \mathbb{F}_2^{2^n}$ given by $X = A \otimes e_1 + B \otimes e_2$ and $X' = B \otimes e_1 + (A+B) \otimes e_2$ where $A = \begin{...
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Orbit of a Permutation

On page 66 of these notes is proposition 4.26: Every permutation can be written (in essentially one way) as a product of disjoint cycles. The proof begins as follows: Let $\sigma \in S_n$, and ...
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1answer
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Prove that $\operatorname{Stab}_G(y) \cong\operatorname{Stab}_G(x).$ [closed]

Suppose that $G$ acts on a set $X$. Let $x,y \in X$ and suppose that $y \in G\cdot x$. Prove that $$\operatorname{Stab}_G(y) \cong\operatorname{Stab}_G(x).$$ Can anyone help me with this ...
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1answer
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Compute a set $S$ given information about how it is acted upon transitively by $D_8$

Let $D_8=D_{2 \cdot 4}$ be the dihedral group on a regular $4$-gon. Suppose that $S$ is a subset of $S_4$, such that S contains the element $( 1 \ 2 \ 3)$. We also know that $D_8$ acts transitively ...
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Can a group act on the empty set?

There isn't much more to add to this question. Can we define an action between some group and the null set? I would have thought that there being no elements to act on it trivially satisfies the ...
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2answers
233 views

Group actions of $D_5$

I have to give $5$ examples of $D_5$ acting on a set. So far, I have $D_5$ acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection ...
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If $b, c \in G$ act on $T$ and fix disjoint (but nonempty) sets of vertices, then $bc$ doesn't fix any vertex

Suppose we have a group $G$ acting on a tree $T$ without inversions (i.e. it is never the case that $g \in G$ flips an edge). Let $b, c \in G$ such that $b$ and $c$ individually fix at least 1 vertex, ...
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1answer
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Proving a mapping is a group action

Let $G$ be a finite group, $S$ be the set of all subsets of $G$ of size $n$, and for $g \in G$, $T \in S$ define $g.T=\{gt: t \in T\}$. My course's notes says that this is a group action of $G$ on $S$...
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27 views

Equivariance theorem on group actions

The following theorem was stated in my algebra class today Theorem: If a group $G$ acts on a set $\Omega$, then for each $x \in \Omega$ we have a $G$-equivariant bijection $f : G/\operatorname{Stab}...
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27 views

Conjugation action on cosets

Given a group $G$ and a subgroup $H < G$, does conjugation given an action on cosets of $H$ via $g \cdot xH = gxHg^{-1}$? If it does it seems there is an easier proof to the problem in Normal ...
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Help with properly discontinuous action implication in Milnor-Schwarz lemma.

From Office Hours with a GGT I am not seeing how proper discontinuity implies there are finitely many translates of $B$ that have distance at most $D$ from $B$. Proper discontinuity says that ...
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1answer
28 views

Non nesting action on Real Trees

Background Let $G$ be a group acting by homeomorphisms on an $\mathbf{R}$-tree $T$. The element $ g \in G $ is elliptic if the fixed point set $\operatorname{Fix}(g)$ is non-empty. The group action ...
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1answer
58 views

Category of G-equivariant sets

I'm getting my very first introduction to categories. Recall that for $G$ a group and $(X, \phi), (X', \phi')$ $G-$sets, a $G$-equivariant map is a map $f : X \longrightarrow Y$ such that for all $g \...
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Character of representation of map from a finite set $M$ to $\mathbb{C}$

Let $G$ a finite group which act on a finite set $M$. Let $C(M) : = Map(M, \mathbb{C}:= \{f: M \rightarrow \mathbb{C} \}$ the vector space of complex values functions from $M$. The group $G$ act on $C(...
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64 views

Cardinality of the set of orbits

Let $G$ be a (possibly infinite) group and $H \unlhd G$. The normality of $H$ in $G$ naturally introduces the $H$-valued map $(g,h) \mapsto g^{-1}hg$, which turns out to be a $G$-action on $H$. Given $...
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1answer
24 views

Question regarding the kernel of $\Lambda:\operatorname{SL}(2)\rightarrow \operatorname{Isom}(\Bbb H_2)$

Let $\Lambda:\operatorname{SL}(2)\rightarrow \operatorname{Isom}(\Bbb H_2)$ be defined by $\Lambda(g)(z)=\frac{g_{(1,1)}z+g_{(1,2)}}{g_{(2,1)}z+g_{(2,2)}},g\in SL(2),z\in \Bbb H_2$. I need to check ...