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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18 views

Quotient of Hausdorff space by free discrete group action is Hausdorff

Let $X$ be a Hausdorff topological space and $G$ a group of homeomorphisms from $X$ to $X$. Suppose that $G$ acts on $X$ discretely, that is, for any $a,b$ in $X$, there exists a neighbourhood $U_x$ ...
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Isometry group of $\mathbb{Z}$

Consider the group of integers $\mathbb{Z}$ equipped with the discrete metric $$d(m, n) = \begin{cases} 1, \quad m \neq n\\ 0 , \quad m = n \end{cases}$$ In particular, $\mathbb{Z}$ is a metric Lie ...
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30 views

If a finite group acts freely, then it acts properly discontinuously.

In my differential geometry course, we saw that a group of diffeomorphism $G$ of a smooth manifold $M$ acts properly discontiuously on $M$ if each $x \in M$ admits an open neighborhood $U$ such that $...
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36 views

Why any permutation defines an automorphism?

My professor kept on saying that: any permutation defines an automorphism. My questions are: 1-Is there a rigor proof for this fact please? I know that a permutation of a set $M$ is a bijective ...
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43 views

how did we know the action in $\mathbb{Z}_{11} \rtimes_{\varphi} \mathbb{Z}_{5}$?

our professor was explaining to us $\mathbb{Z}_{11} \rtimes_{\varphi} \mathbb{Z}_{5}$ and he gave us the following automorphism: $\varphi: \mathbb{Z}_{11} \rightarrow \mathbb{Z}_{11} $ defined by $x \...
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1answer
34 views

The Variety of 3-dimensional Real Lie Algebra Structures

Let $C^{2}(\mathbb{R}^{3};\mathbb{R}^{3})$ be denote the vector space of all skew-symmetric bilinear maps from $\mathbb{R}^{3}\times\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ and let $\operatorname{Lie}(\...
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24 views

Lift a map between quotient manifolds

Let $M$ and $N$ be two smooth manifolds. Assume that a compact group $G$ acts smoothly, freely and properly on both $M$ and $N$. Then any $G$-equivariant map $f:M\rightarrow N$ induces a map $\...
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1answer
25 views

elaboration of group action definition.

My professor wrote at the beginning of speaking about group actions this: In general, Aut$(X) \subset $ Sym$(X)$ acts on $X$. If $G \subset Aut(X)$ is a subgroup, we say that "G acts on $X$ by ...
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1answer
59 views

Trivial group action

On page 43 of Dummit & Foote's abstract algebra: Let $G$ be a group and $A$ a nonempty set. Let $ga = a$, for all $g \in G$, $a \in A$. This action is called the trivial action and $G$ is said to ...
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Plane tiling with discrete group of positive isometries

Let $G$ be a group of positive isometries of the plane (and hence $G$ acts on the plane). Suppose that any orbit of $G$ is discrete and let $P=\{x\in \mathbb{R}^2: \|x-a\|\leq \|x-g(a)\| \text{ for ...
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45 views

Dummit Foote Exercise 1.7.10.a permutations of subsets self-study

Problem Statement: Let $A$ be a nonempty set and let $k$ be a positive integer with $k\leq\left|A\right|$. The symmetric group $S_A$ acts on the set $B$ consisting of all subsets of $A$ of cardinality ...
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Split a regular tetrahedron into four equal сonvex polyhedrons.

A point is marked on the face of a regular tetrahedron. Prove that the tetrahedron can be split into four equal сonvex polyhedrons in a way that this point is a vertex of one of them. This problem is ...
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20 views

Taking fixed points does not commute with the tensor product

In the first chapter of his book on Class Field Theory, Neukirch mentions that for a group $G$ and two $\mathbb Z [G]$-modules $A$ and $B$, the tensor product $A^G \otimes_{\mathbb Z} B^G$ is in ...
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How can a set of fixed points be viewed as a $N_{G}(H)/H$ - set?

Okay, so let $G$ be a group and $H$ - its subgroup. Moreover, let $X$ be a $G$-set. Let's consider a set of fixed points of $H$ on $X$ and let's denote it by $X^{H}$. The author of the paper I'm ...
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33 views

Finding the orbit of an element in $S_5$

I'm confused on how to find the orbits of elements. The orbit for $\sigma\in S_X $ is defined to be $O_{x,\sigma} = \{y\,|\,\sigma^n (x) = y\}$. I need to compute the orbit of the permutation $\alpha =...
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56 views

About groups act faithfully on a set

Suppose group $G$ act faithfully on a set $X$ of $5$ elements, and there are $2$ orbits, of order $2$ and $3$ respectively. Then what should the group $G$ be like? Note: A group $G$ acts faithfully on ...
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58 views

Classify all maps $f: (\mathbb{R}^d)^n \to \mathbb{R}^d$ such that $f(R x_1,\dotsc, R x_n) = R f(x_1,\dotsc, x_n)$ for all $R \in O(d)$

Let $O(d)$ denote the orthogonal group for $\mathbb{R}^d$ with the standard Euclidean inner-product. A nautral question is to find all possible functions $f: \mathbb{R}^d \to \mathbb{R}$ that are ...
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1answer
22 views

Semigroup action generalized (relations instead of functions)

If with every element of a (semi)group is associated a function, it is basically called a (semi)group action. What if with each element of a semigroup is associated a relation? That is, formally, what ...
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1answer
56 views

Confusion on how groups acts on objects

im trying to learn group theory to understand particle physics and im currently reading: "A Simple Introduction to Particle Physics ,Part I - Foundations and the Standard Model". In the ...
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Sharply $k$-transitive actions on spheres

A nice fact from complex analysis is that the mobius group acts sharply 3-transitively on the Riemann sphere. I am wondering if other sharply k-transitive (continuous) actions are known on any $S^n$, ...
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1answer
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Find all matrices that belong to the stabilizer of a specific vector.

I am working on this problem and not sure about my solution so I would like some help: Find all matrices in the group $\operatorname{GL}_2(\Bbb{F}_5)$ that are in the stabilizer of $\begin{pmatrix} 0\...
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How units of a Lie groupoid act on a manifold?

The definition of a Lie groupoid action given in Sébastien Racanière's notes (up to notation) says that the action of a Lie groupoid $\mathcal{G} \rightrightarrows M$ on a smooth manifold $Q$ consists ...
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23 views

About the splitting of tangent space in the neighborhood of a point

Let $G$ be a Lie group, let $x$ be an arbitrary point in $G$ and let $\mu: G \times M \to M$ be a free proper action of $G$ on a smooth manifold $M$. Since $\mu$ is free we can conclude that the image ...
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30 views

What are the fixed points of $S = \{A, B,C,D,E \}$ under the action of $D_4\ ?$

Let $S = \{A,B,C.D,E \}$ be the points located on the square in Figure $2.$ Let $D_4 = \{1,r,r^2,r^3, s, rs, r^2 s , r^3 s\ |\ r^4 = 1, s^2 = 1, rs = sr^{-1} \}$ be the dihedral group, where $r$ is ...
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24 views

Thus $O[G$ \ $ H]$ is projective as an $O(P × P)$-module.

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup and $H$ a subgroup of $G$ containing $P$ such that $P ∩ {^xP} =$ {$1$} for all $x ∈ G$\ $H$. $O$ is a complete DVR. The action by $P × P$ on $G$ \ $...
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1answer
48 views

Count the number of orbits under the action of $S_4$ on $\mathcal P (X),$ where $X = \{1,2,3,4 \}.$

Let the symmetric group $S_4$ act on $X = \{1,2,3,4 \}.$ This gives an action on the power set $\mathcal P(X).$ Count the number of orbits for the action of $S_4$ on $\mathcal P(X).$ My attempt $:$ ...
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31 views

Prove $G$ acts $(k + 1)$-transitively on $A$ $\iff$ $\exists a \in A, G_{(\{a\})}$ acts $k$-transitively on $A \setminus\{ a \}$

Definitions The pointwize stabiliser $G_{(A)}$ is defined as the set $\{g \in G | \forall a \in A: g(a) = a \}$ A subgroup $G ≤ Sym(X)$ acts $k$-transitively on a subset $A ⊆ X$ if $|A| ≥ k$ and $\...
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1answer
87 views

Orbits of $\mathbb{Z}$ action on $\mathbb{S}^1$

Let $\mathbb{Z}$ acts on $\mathbb{S}^1$ with the action $\phi (n)(z)=e^{i \alpha n}z$ defined for $\alpha\in\mathbb{R}$ and $z\in\mathbb{S}^1$. First we easily see that $\phi$ is stable on $\mathbb{S}^...
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17 views

Bounded-to-1 factors preserve topological entropy for amenable group action

I have seen a proof of the following result (about dynamical systems with an action by the integers): Given a factor map $\phi:\,(X,T)\rightarrow(Y,S)$ between two topological dynamical systems. If $\...
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1answer
32 views

Group acting properly and cocompactly by isometries on a metric space has finitely many conjugacy classes of point stabilizers

Let $\Gamma$ be a group acting properly and cocompactly by isometries on a metric space $X$. Then, $\Gamma$ has finitely many conjugacy classes of point stabilizers. The proof of this well-known ...
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1answer
54 views

Group acting coprimely by automorphism

Consider that a group $A$ acts by automorphism on a finite group $G$. If this action is coprime, i.e. $\gcd(|A|,|G|)=1,$ can we affirm that this action is fixed point free, i.e. $C_G(A)=1$? I tried to ...
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1answer
38 views

Use Burnside's theorem to find the colourings of an octahedron

The question is to find in how many ways we can colour the edges of an octahedron with $k$ colours by using Burnside's theorem. I already know that I'm supposed to find the automorphism group to get $|...
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2answers
57 views

establish the diffeomorphism between $S^2$ and $SO(4)/U(2)$

I was trying to do this using the idea of homogeneous space, but stuck on the point of how $SO(4)$ acts on $S^2$. Naturally $SO(3)$ acts on $S^2$ transitively, but how does the group action works for $...
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A question about Hausdorff quotient by Group Action. Also, there are some very related material in Lee's book.

Let $G$ be a group acting on a topological space $X.$ I wanted to check the following statement. "Recall that the action is called even (or classically properly discontinuous) if for every point ...
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1answer
225 views

Why is this quotient of the punctured plane not Hausdorff (Hatcher 1.3.25)?

This is from question 1.3.25 of Hatcher's Algebraic Topology: Let $\phi : \mathbb{R}^2 \to \mathbb{R}^2$ be the linear transformation $\phi(x,y) = (2x, y/2)$. This generates an action of $\mathbb{Z}$ ...
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1answer
32 views

Automorphism of $\mathbb{H}$ uniquely determined by a point of $\mathbb{H}$ and a point of $\partial \mathbb{H}$

I'm reading some lecture notes on complex analysis, where the following is stated: An element of $\textrm{Aut}(\mathbb{H})$ is uniquely determined by the image of a point in $\mathbb{H}$ and of a ...
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44 views

GAP computational group action question

Let $G$ be a group generated by some permutations $G = \langle A,B,C,D,E \rangle$. Suppose $G$ acts on a set $X$. Given some subset $X' \subseteq X$ and an injective map $f : X' \to X$, How can I ...
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2answers
125 views

Group Action notation and understanding

I've been studying Group Theory for some time, but I still do not understand the appeal of the group action notation. I mean - every time I see: $a \cdot e$, I am thinking, why not just treat $a$ as ...
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1answer
78 views

Understanding why proving a hint will prove that $D_{6} \cong S_{3} \times \mathbb{Z}_{2}.$

Recall that the dihedral group $D_{6} \cong \mathbb{Z}_{6} \rtimes _{\phi} \mathbb{Z}_{2},$ where the reflection $\mu$ acts on the rotation $\rho$ by \begin{equation*} \prescript{\mu}{}{\rho} = \...
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41 views

Understanding how reflection acts on rotation in $D_{6}.$

Here is what my professor wrote: Recall that the dihedral group $D_{6} \cong \mathbb{Z}_{6} \rtimes _{\phi} \mathbb{Z}_{2},$ where the reflection $\mu$ acts on the rotation $\rho$ by \begin{equation*} ...
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28 views

Lagrangian grassmannian of all lagrangian subspaces in $\mathbb{R}^n\times \mathbb{R}^n$ can be identified with $U(n,\mathbb{C})/O(n,\mathbb{R})$

I've been trying to proof this using $U(n,\mathbb{C})$ action over all lagrangian subspaces of $\mathbb{R}^n\times \mathbb{R}^n$ but it didn't work. I mean, I got stuck and I didn't know what else to ...
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64 views

Some questions about the orbit of a scheme under action of a group scheme

I'm reading Geometric Invariant Theory by Mumford and get confused about statements on Page 6-7.enter image description here I'm confused about the meaning of ii) for every algebraically closed ...
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49 views

Transitive subgroup of $S_n$ that has order n and consists of derangement.

Besides $Z_n$, is there any subgroup of $S_n$ that has order n and the group actions are all derangements?
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56 views

Orbits and stabilizers of the permutation group.

Let $G$ be a group s.t $G = \langle(12),(345)\rangle \subseteq S_5$ acts on the set $X = \{1,2,3,4,5\}$. I want to find all orbits and stabilizers of $G$. The point I don't understand is that ...
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1answer
64 views

Prove that a map is one-to-one if $G$ is finite group and $H$ and $K$ are its subgroups

Okay, so I've been struggling with one problem lately. I know it must be something very obvious and the solution is right in front of me, but I just can't see it. Let's get to the problem: Let $G$ be ...
2
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1answer
35 views

Group Action from to Subgroup to Group: Care in Differentiating Between Binary Op., Group Action Op.

Using Dummit & Foote in a graduate level abstract algebra class, and it is my first introduction to group actions. I'm getting a better understanding as I go along, but one thing still puzzles me: ...
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5answers
70 views

general linear group $GL_{2}(\mathbb{Z}_3).$ [closed]

finding the order of the general linear group $GL_{2}(\mathbb{Z}_3).$ ? Is finding the order by listing its elements only ? or is there a smarter method?
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1answer
98 views

Prove *by group actions* that there doesn't exist any normal subgroup $H$ such that $S_5/H $ is isomorphic to $S_4$

I was trying to give an answer to this very same question by means of group actions (so I think this is not a duplicate, at least as long as the answers therein do not use group actions). My attempt: ...
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21 views

How do I engineer the equivalence of graph-connectedness of an orbit and topological space connectedness?

Let $X,d$ be Hausdorff and let the level sets of a function $f:X\to X$ converge to their image. Is the directed graph of the orbit of $f$ in $X$ graph-connected if and only if $X,d$ is connected? ...
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54 views

Is there any difference between $X\times^G Y$ and $X\times_G Y$?

Let $X,Y$ be "spaces" on which a group $G$ acts. Typically I see the notation $X\times_G Y$ to denote the quotient of $X\times Y$ by the relation $(x,y) = (gx,gy)$. In a paper I now see the ...

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