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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25
votes
0answers
322 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 ...
17
votes
0answers
393 views

Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la Harpe,...
10
votes
0answers
369 views

Distributing groups of objects into boxes

How can I enumerate the number of ways of distributing distinct groups of identical objects (but various cardinality) into $k$ boxes such that at most one box is empty $(1)$ and no combination of ...
9
votes
0answers
190 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 ...
7
votes
0answers
259 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 ...
6
votes
0answers
51 views

Proving that the given map has the path lifting property

Let $X$ be a locally compact metric space and $G$ be a discontinuous group of homeomorphisms of $X$. I need to show that the orbit map $p :$ $X \rightarrow X/G$ has the path lifting property. ...
6
votes
0answers
55 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
189 views

Normal subgroup $H$ of $G$ with same orbits of action on $X$.

I have a somewhat broad question related to group actions and their restriction to a normal subgroup. If we have a group action $\sigma : G \times X \rightarrow X$ with orbits $G_x$, and a normal ...
5
votes
0answers
81 views

What are some topos-theoretic insights about $G$-sets?

Since a $G$-set is just a functor $G\longrightarrow \mathsf{Set}$, the category of $G$-sets seems to be a simple example of a topos. What are some topos-theoretic insights into $G$-sets? Insights ...
5
votes
0answers
187 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 ...
5
votes
0answers
124 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) = g^{-1}f(p)...
5
votes
0answers
430 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 ...
4
votes
0answers
48 views

Explicit Dehn twist for $S^n\times S^n$

Fix $n$ odd and let $M=S^n\times S^n$. The diffeomorphism group Diff($M$) acts on the homology group $H_n(M)\simeq \mathbb Z^2$ inducing a surjection $d: \text{Diff}(M) \rightarrow \text{SL}(2,\mathbb ...
4
votes
0answers
61 views

group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
4
votes
0answers
64 views

Is $D$ a metric on $X/G$ and does it induce the quotient topology?

Let $(X,d)$ be a compact metric space and $G$ be a finite group of homeomorphisms of $X$. Let $p:X\rightarrow X/G$ be the orbit map. Then we can define a (psuedo) metric on $X/G$ as follows - $$D(\...
4
votes
0answers
62 views

Can we extend the group action from subgroups?

Let $G$ be a group and $H,K$ be subgroups of $G$ such that $G=<H,K>$. Suppose that $H,K$ acts on the set $S$. Is there any condition that which guarantees that action of $H,K$ is extended to ...
4
votes
0answers
46 views

Topology on $\text{Homeo}(X)$ Which Captures Topological Group Actions.

Definition. Let $G$ be a group and $X$ be any set. We may define a group action of $G$ on $X$ as map $\cdot: G\times X\to X$ such that $e\cdot x=x$ for all $x\in X$ and $g\cdot(h\cdot x)=gh\cdot x$ ...
4
votes
0answers
59 views

Elementary consequences of commuting limits and colimits over groups

In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is mentioned: Theorem 1. $H$-limits commute ...
4
votes
0answers
189 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
95 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
442 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
83 views

Group action and orbit space

Suppose some group, G, acts on a space, X. Then an orbit of some $x\in X$ is defined as $$G.x = \lbrace g.x \mid g\in G\rbrace$$ Now consider the orbit space, $X/G$, the set of all orbits. I'm finding ...
3
votes
0answers
50 views

How to describe the points of a quotient stack?

Let $G$ be a finite algebraic group acting on a projective complex variety $X$. Then a quotient $Y=X/G$ exists as a scheme and, if $G$ acts freely, $Y$ is an orbit space and the natural map $$\eta:[X/...
3
votes
0answers
64 views

Group actions by semi-direct products of groups

I have trouble to understand the second part of the following example which I hope someone can explain to me. First let me explain the initial situation which I feel comfortable with: Consider the ...
3
votes
0answers
48 views

Moment map in general

Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if ...
3
votes
0answers
49 views

Free and proper action of a closed subgroup of a Lie group

I'm taking a course on Riemannian geometry and in my homework set I'm asked to prove that the (left) action of a closed subgroup $H$ of a Lie group $G$ on $G$ is free and proper. To prove that it is ...
3
votes
0answers
42 views

A problem of a discrete group of smooth isometries acting discontinuously on a smooth manifold.

Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly ...
3
votes
0answers
47 views

Verify proof to show this is a $\sigma$ - locally finite basis.

Can someone tell me if what I have done is correct? Proposition : Let $X$ be a compact metric space and $G$ a finite group acting on it. Let $p : X \rightarrow X/G$ be the orbit map. Let $B = \{ ...
3
votes
0answers
83 views

Reference Request: Group Theory via the Group Action Perspective

I am looking for a higher undergraduate or graduate level textbook that introduces group actions after groups just as many textbooks introduce modules after rings. I think the semigroup/semigroup ...
3
votes
0answers
37 views

Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
3
votes
0answers
32 views

Transfer homomorphism in transformation groups

I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups. But suppose ...
3
votes
0answers
168 views

Homogeneous metric on a homogeneous space $G/K$ - is this the same as a $G$ - invariant metric?

I have trouble putting down the notion of a homogeneous Riemannian metric. Suppose we are given a Riemannian manifold $(M,g)$ on which a compact Lie group $G$ acts transitively by isometries (this ...
3
votes
0answers
42 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
3
votes
0answers
80 views

Motivation behind automorphism bases?

Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = 1_\mathcal{...
3
votes
0answers
57 views

Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
3
votes
0answers
113 views

Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
3
votes
0answers
167 views

Orbits of the group action $g. (x,y) = (gx,gy)$ on cartesian product

Let $G$ be a group acting on a set $X$. Then we have a natural action on $X \times X$ in the following way: $g. (x,y) = (gx,gy)$. Then, suppose we have two points of interest $x_1,x_2 \in X$, and we ...
3
votes
0answers
52 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 ...
3
votes
0answers
34 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
75 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
82 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 ...
3
votes
0answers
430 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 $|A|\ge2.$ ...
3
votes
0answers
180 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
37 views

Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. ...
2
votes
0answers
35 views

Group action with two normal subgroups which induce same block system

So awhile back I asked this question here on stack exchange: Normal subgroup $H$ of $G$ with same orbits of action on $X$. At the time I wasn't quite sure what I was really wanting to know about ...
2
votes
0answers
41 views

Do finite groups act admissibly on separated schemes of finite type over k

Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$...
2
votes
0answers
20 views

Is $Y/K$ homeomorphic to $Y'$ as defined below -

Let $G$ be a topological group acting on a topological space $X$ in such a way that there are only finitely many orbits. We will fix points $x_1,\cdots,x_n\in X$ and let $X=\bigcup_{i=1}^n G\cdot x_i$ ...
2
votes
0answers
35 views

Fixed point set as an inverse limit

We can regard a group action on a set as a functor $$ F: BG \to Set\;, $$ where $BG$ is the category with one object and a morphism for each element of $G$, and $Set$ the category of sets. Now, is ...
2
votes
0answers
44 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
2
votes
0answers
86 views

Dense and turbulent orbits

In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition ...