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 $2n$ acts 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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votes
2answers
53 views
Group and orbit question.
Suppose group $G$ acts on a set $A$.
a) If $x$ and $y$ are in the same orbit, show that there exists some $g \in G$ such that $gG_x g^{-1} = G_y$.
b) Show that if $|G.x|$ is finite, then $|G.x| = ...
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votes
2answers
66 views
Composite group homomorphism between alternating groups
Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
4
votes
1answer
163 views
Quotient of a locally compact Hausdorff space by a proper action is Hausdorff
I am trying to prove the following:
Let $G$ be a topological group acting properly on a Hausdorff locally
compact space $X$, i.e. preimages of compacts sets by the map
$$G\times X\to X\times ...
1
vote
1answer
85 views
Stabilizer of a 4 by 4 skew symmetric matrix by orthogonal matrix
Matrices are over the field of complex numbers, and $X^t$ means transpose of a matrix $X$.
Consider the group action of $O(4)=\{P\mid PP^t=I\}$ on $SK(4)=\{M\mid M^t=-M\}$ by $(P,M) \rightarrow ...
6
votes
0answers
34 views
Chern classes of free quotient manoflds
Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
4
votes
0answers
61 views
The classifying space of a gauge group
Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
4
votes
0answers
147 views
Semi-direct product isomorphic to direct product
I would like some help on the following problem from anyone who would like to help.
Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$.
The situation ...
2
votes
0answers
54 views
Group actions and associated bundles
Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
2
votes
0answers
37 views
Semi-orbital equivalence relation
Edit: I was in kind of a hurry when writing this post and made a mistake in the formula defining $G_E$. What I had written said that $G_E$ preserves the set of classes of $E$, while I meant actually ...
2
votes
0answers
72 views
Group action and Radon measure
Let $\mathscr M(\mathbb R)$ be the Banach space of complex-valued Radon measures on $\mathbb R$, and let $\pi$ be the action of $\mathbb R$ on $\mathscr M(\mathbb R)$. Let $\mathscr A$ denote a subset ...
2
votes
0answers
46 views
Infinitesimal generators of actions
Is there a method to obtain an action of an infinite dimensional Lie group starting with its infinitesimal generator ?
I'm interested about actions of G on itself . And I was wondering if I can ...
2
votes
0answers
87 views
Properly discontinuous action on a non-locally compact space
Let me begin with some definitions in order to avoid confusion.
An action of a group $G$ on a space $X$ is proper if the map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, gx)$ is proper, ...
2
votes
0answers
130 views
Semidirect product group actions
$H$ and $K$ are groups, and $\Gamma$ is a set acted upon by H, while $\Delta$ is a set acted upon by $K$. Let $W := K \wr_\Gamma H$, the wreath product of $H$ and $K$. I have seen theorems stating ...
1
vote
0answers
29 views
Action of a Lie group on a coset of its subgroup
I am a physicist, so sorry for the lack of rigor. It is well known that a (say compact) Lie group $G$ acts naturally by left multiplication on the coset space $G/H$ where $H\subset G$ is its (Lie) ...
1
vote
0answers
26 views
Orbits of the action of G/H
Let $G \subset Iso(M)$ be a Lie group which acts on a (semiriemannian) manifold $M$ properly and smoothly. Let we know the orbits of the action. Suppose that $H$ is a discrete central subgroup of $G$ ...
1
vote
0answers
57 views
Isomorphism between G-equivariant bijections and Normalizer
If anyne could give me some help with this, it will be deeply appreciated:
Let $X$ be a set equipped with the action of some group $G$. Denote by $Aut_G(X)$ the set of $G-equivariant$ bijections $f: ...
1
vote
0answers
36 views
A K-invariant submanifold of G-manifold and fundamental vector fields
Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$
$$
\underline{X}(p) := ...
0
votes
0answers
41 views
Definition of g-orbit of a set
Let $g$ be a Lie algebra and $M$ a manifold, what does mean $g$-orbit of $M$?
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votes
0answers
25 views
Finiteness of fixed points of a Lie group action
Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$.
Under which assumptions there will be a ...
0
votes
0answers
26 views
What is the Lobachevsky space?
I am reading about Lie group actions on manifolds and the author used the Lobachevsky space $SO^{+}_{1,n}/SO_n$ with the actions of the subgroups $SO_n$, $SO_{1,n-1}$ and the horispherical group ...
0
votes
0answers
60 views
Actions of G on corresponding orbits are equivalent for stable maps
Given actions of G on X and on Y, these actions are equivalent if and only if there is a bijection from $X $ \ $ G \rightarrow Y$ \ $G$ so that actions of G on corresponding orbits are equivalent.
...
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votes
0answers
77 views
Questions (doubts) on: Group Action on Manifolds
There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up:
Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...
0
votes
0answers
44 views
G-H biset and left & right G-set
1) Prove that each left $G$-set $X$ can be turned into a right $G$-set by defining $x\sigma = \sigma^{-1}x$, and that every right $G$-set arises in this way.
I have the left action $X \times G \to X$ ...
