# Tagged Questions

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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### 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 ...
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### 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,...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $g\cdot(h\cdot x)=gh\cdot x$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$\eta:[X/... 0answers 64 views ### Group actions by semi-direct products of groups I have trouble to understand the second part of the following example which I hope someone can explain to me. First let me explain the initial situation which I feel comfortable with: Consider the ... 0answers 48 views ### Moment map in general Let the Lie group G act on the smooth manifold X with the map (g,x)\to gx. In any point x\in X, the differential of this map induces a linear map:$$ \mu:T_e G \to T_xX\;, $$and globally, if ... 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 ... 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 ... 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 = \{ ... 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 ... 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 ... 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 ... 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 ... 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 ... 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{... 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 ... 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 ... 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 ... 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 ... 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 \... 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 ... 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 ... 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. ... 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 ... 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. ... 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 ... 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... 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 ... 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 ...
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 $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...