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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13
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0answers
131 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
8
votes
0answers
144 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
6
votes
0answers
39 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 ...
5
votes
0answers
121 views

Why are they called orbits?

When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\} $$ that are called orbits. Although, the only reason I find convincing for that name is that in ...
5
votes
0answers
96 views

Group action on a manifold with finitely many orbits

I'm looking for a result along the lines of the following: Let $G$ be a group acting on a set $X$. If the action partitions $X$ into finitely many $G$-orbits, then $\dim G \geq \dim X$. For ...
4
votes
0answers
100 views

Fundamental group of a space under a group action

The short version: Why does a Borcea-Voisin threefold has trivial fundamental group? Well, what is a Borcea-Voisin threefold? These are named after Borcea and Voisin, who introduced a method of ...
4
votes
0answers
69 views

Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
4
votes
0answers
163 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 ...
4
votes
0answers
86 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
226 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 ...
3
votes
0answers
28 views

Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
3
votes
0answers
52 views

Dimension of a constructible set intersecting each orbit of a $G$-variety

In preparing a talk I'm having trouble with exercise 3 and 4 on page 25 of the following Lecture Notes of Crawley-Boevey (I only need the case $X=Y$ there): $\text{3.}$ Let $X$ be a variety ...
3
votes
0answers
99 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
2
votes
0answers
36 views

Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
2
votes
0answers
11 views

Finding a fundamental polygon for two-generator subgroup of PSL(2,R)

Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like ...
2
votes
0answers
41 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
2
votes
0answers
88 views

Stabilizer map on transitive G-set defines a morphism with G acting on subgroups by conjugation

This is part of a homework problem for a graduate course on abstract algebra. Given a transitive G-set $X$, show that the map that assigns to $x \in X$ its stabilizer defines a morphism of G-sets ...
2
votes
0answers
171 views

Prove that a doubly transitive group is primitive.

My Attempt: (a) $S_n$ is transitive on $\{1,2,\dots,n\}$ and for any $(i,j)\in G_a,~(i,j)i=j$ whence $G_a$ is transitive on $\{1,2,\dots,n\}-\{a\}.$ (b) Without loss of generality let $|A|\ge2.$ ...
2
votes
0answers
99 views

Degree of factor in a resolvent

Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma ...
2
votes
0answers
83 views

trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...
2
votes
0answers
70 views

An orbit of a group action and the implicit function theorem

Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
2
votes
0answers
87 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
115 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
173 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
206 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
34 views

Ring A is integral over the subring of invariants under a finite group action

I need to prove that if G is a finite group that acts on ring A, and $A^G$ is the subring consisting of elements of A which are invariant under all g in G, then A is integral over $A^G$. The hint ...
1
vote
0answers
17 views

Circle action on the product of a Mobius band and a circle.

Consider the product of a Möbius band and a circle $Mo\times S^1$. Is there a circle action on $Mo\times S^1$ such that it is equivariantly homeomorphic to the twisted product $D^2 ...
1
vote
0answers
24 views

Notation for pointwise versus “setwise” stabilizers

Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ...
1
vote
0answers
37 views

How we show primitive action shows alternating group

I have a graph (as shown in figure), which represents a quotient of the group $$G=\langle A,B,C,D; A^3=B^2=C^3=D^2=(AC)^2=(AD)^2=(BC)^2=(BD)^2=1 \rangle.$$ I proved that $G$ acts 2-transitively and so ...
1
vote
0answers
30 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...
1
vote
0answers
37 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
1
vote
0answers
48 views

Find the orbit space $T^2 / \mathbb Z_2$

Let $T^2$ be the unit torus $$ T^2 = \left\{ (\lambda, \lambda') \in \mathbb C^2 \mid |\lambda| = |\lambda'| = 1 \right\}. $$ Then the group $\mathbb Z_2$ is acting on $T^2$ by the rule ...
1
vote
0answers
63 views

Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
1
vote
0answers
59 views

Tangent bundle of a quotient manifold

I am interested in the tangent bundle of the quotient of a manifold by a proper and free action but I can't find any reference on the net. Does anyone know a book or article where it is described ?
1
vote
0answers
39 views

When the $(\mathbb R,+)$-action defined by the flow of a vector field on a manifold is proper ?

I'm interested in the conditions that a vector field has to satisfy in order for its flow to define a proper action. Technically, let $\xi$ be a smooth vector field. For each $m\in M$, there is a ...
1
vote
0answers
42 views

Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
1
vote
0answers
50 views

Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
1
vote
0answers
45 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
38 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 ...
1
vote
0answers
72 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. ...
1
vote
0answers
77 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
47 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
21 views

How many orbits are there in the group action of S_4 on X={1,2,3,4}?

So far I know that the order of the group is 4! which is 24 elements. I know that X the set has 4 elements. Finally, I understand that the Orbit-Stabilizer theorem states that: ...
0
votes
0answers
31 views

Burnside's Lemma and Stirling Numbers of the First Kind

I've seen that $n!=\displaystyle\sum_{p=0}^n s(n, p)n^p$, where $s(n, p)$ are the signed Stirling Numbers of the First Kind, whose absolute values count the number of permutations in $S_n$ which have ...
0
votes
0answers
23 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
0
votes
0answers
28 views

Finding unique manifold structure by action

Let $M$ be a manifold and suppose that the discrete group $G$ acts smooth and faithfull on $M$. Assume that we know that the orbit space $M/G$ is Hausdorff with respect to the quotienttopology and ...
0
votes
0answers
33 views

Orbits of action of $SL_2(\mathbb{Z})$ on lattice

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$. Specifically, what are the orbits of this action?
0
votes
0answers
39 views

Can we prove $H \cong xHx^{-1}$ given $H \le G, x \in G$ using group action?

The exercise is as follows: $G$ is a group, $H \le G$. For any $x \in G$, to prove that $H \cong xHx^{-1}$. I am able to prove this isomorphism by defining a bijection $f : h \mapsto xhx^{-1}$ ...
0
votes
0answers
7 views

$\bar{L}$ points of $GL_{(ab)^2}/PGL_a\times PGL_b$

I am reading the paper "Matrix invariants of composite size" by A. Schofield (see here), and I have trouble understanding some one of his arguments, and I hope someone can explain them to me. He ...
0
votes
0answers
17 views

How to write Cayley representation permutations as cycles?

The representation of $g \rightarrow xg$ was given to us as Cayley representation. I believe it means that every group element $g \in G$ is mapped to another element $xg$ where $x$ is the same for ...