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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27 views

Determining the corresponding vector field to a group action.

Im having trouble trying to understand how to determine the corresponding vector field to a group action on a symplectic manifold. I feel this will be easier if I give two examples which are confusing ...
0
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1answer
22 views

points which are fixed points of a finite group action

consider an open set $\tilde{U}\subset\mathbb{R}^n$ and a finite Lie-group $G$, which acts smoothly on $\tilde{U}$, i.e. we have a smooth map $G\times \tilde{U}\rightarrow\tilde{U}$. Suppose further, ...
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0answers
20 views

Does the Weyl group act on its Lie algebra?

I am trying to prove something about the action of a particular Lie algebra on a particular representation (it's the starred claim on page 7 of this paper for those interested). My friend showed me a ...
0
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1answer
42 views

Normality of a covering space.

I am having some trouble trying to do problem 1.3.24 in Hatcher's Algebraic Topology: Given a covering space action of a group G on a path-connected, locally path-connected space X, then each ...
1
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1answer
11 views

transitive action on finite abelian subgroups

Let G be a group and K a finite subgroup of G. Let H be some subgroup of the normalizer of K in G, and assume the action of H on K by conjugation is transitive on elements of K of same order. Does H ...
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1answer
26 views

transitive action on finite abelian subgroups [closed]

Let $G$ be a group and $K$ a finite subgroup of $G$. Let $H$ be some subgroup of the normalizer of $K$ in $G$, and assume the action of $H$ on $K$ by conjugation is transitive on elements of $G$ of ...
2
votes
1answer
33 views

Orbit closures of real symmetric bilinear form

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^n)$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$. I ...
5
votes
1answer
76 views

If groups $G$ and $H$ act on $X$, does $G\times H$ act on $X$?

Suppose two groups $G$ and $H$ act on a set $X$. What is the a group action of $G\times H$ on $X$? From the actions there a homomorphisms $\varphi\colon G\to S_X$ and $\psi\colon H\to S_X$. So this ...
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0answers
19 views

Orbit closures of symmetric bilinear form

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: ...
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0answers
33 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
3
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0answers
60 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
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1answer
34 views

Group $G$ acting on $\Omega$ such that each $\alpha \in \Omega$ has unique $p$-element fixing $\alpha$.

Let $G$ be a group acting on a set $\Omega$ and let $p$ be a prime. Suppose that for each $\alpha \in\Omega$ there is a $p$-element $x \in G$ such that $\alpha$ is the only point fixed by $x$. If ...
2
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2answers
28 views

Transitive action of $SL_2(\mathbb{R})$ on $\mathbb{H}$

I'm studying Modular Forms and I'm not understanding why the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ defined by $\begin{pmatrix} a & b \\ c & d \end{pmatrix}z=\frac{az+b}{cd+d}$ is ...
0
votes
1answer
25 views

What is a permutation representation in regard to group actions

I have read the definition of a permutation representation from Dummit and Foote, and Wiki, but I don't understand. Can I please have an example? I get the impression that we can write a group action ...
1
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1answer
31 views

Two group elements induce the same permutation on $A$ if and only if they are in the same coset of the kernel.

Page 113 - Dummit and Foote - Group actions Two group elements induce the same permutation on $A$ if and only if they are in the same coset of the kernel. What does this mean? Two ...
2
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2answers
50 views

Homogeneous Isotropic Riemannian Manifolds

In John Lee's book Riemannian Manifolds on page 33, Lee writes "Clearly a homogeneous Riemannian manifold that is isotropic at one point is isotropic at every point". It seems that he means that he ...
2
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0answers
19 views

Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
2
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2answers
47 views

Tree of groups $(\mathcal{G},T)$

Let $G$ be act on $\Gamma$ with a fundamental domain $T$ where $T$ is tree. We construct tree of groups $(\mathcal{G},T)$ with the following structure: $$\text{for every } v\in ...
4
votes
1answer
45 views

Least dimension of Lie group acting transitively of a manifold

I guess that the least possible dimension of a Lie group $G$ acting smoothly and transitively on a compact manifold $M$ is $\operatorname{dim}(M)$. Is this correct and is there a ref?
0
votes
1answer
46 views

Why does it matter that the group of rotations act *freely* in Tao's proof of the Hausdorff paradox?

Consider the following extract from this expository article: (This is a key step in proving the Banach-Tarski theorem.) My question is: if the action of $G$ on $S^2 - C$ were not free, would the ...
1
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1answer
31 views

Set stabilizer of subset vs. set stabilizer of inverse subset

Let $G$ be a finite group and $A\subseteq G$ a subset. The left regular action of $G$ on itself induces a natural action on the powerset of $G$: $$G\times 2^G\rightarrow 2^G,(g,A)\longmapsto ...
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votes
1answer
81 views

Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?

Here $G$-sets denote the category of sets which have a left $G$-action. So the question is whether a functor $F \colon \text{$G$-sets} \to \text{$H$-sets}$ implies that we have an isomorphism of ...
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votes
0answers
32 views

Are permutation group block only defined in the context of finite sets?

From Dummit and Foote (emphasis mine): Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ s.t. $\forall \sigma\in G:$ either $\sigma(B)\cap B ...
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1answer
31 views

Where do I use that $G$ is a permuation group?

This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question. The question is (summarised a bit): Let $G$ be a transitive permutation group on a finite set $A$. ...
3
votes
1answer
38 views

Difference between “$G$ acts on $A$” and “G is a permutation group on $A$ (i.e. $G\leq S_A$)”

This question is inspired by questions $4.1.1$ and $4.1.2$ of Dummit and Foote. The hypothesis for the first question is formulated as: "Let $G$ act on the set $A$", and the hypothesis for the second ...
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0answers
17 views

Are there any resources on this notion of a “directed” semigroup?

Given two additive semigroups $G$ and $H$, and a semigroup action $ \& : G \times H \to G$, then intuitively $H$ can be thought of as a set of "directions" in which we can move in the space $G$. ...
3
votes
1answer
36 views

Level set as the orbit of the action of a Lie Group?

I'm wondering the following. Given a smooth function $f:\mathbb R^n\rightarrow \mathbb R^m$ with $m<n$ and level sets $\mathcal O(y)=\{x\in\mathbb R^n| f(x)=y \}$. What are the conditions on $f$ ...
1
vote
2answers
31 views

Question about action on groups in Bourbaki (Algebra I)

In Bourbaki, Algebra I, chapter I, §5 "Groups operating on a set" paragraph 1, Bourbaki defines the operation of a group $G$ on a set $E$ as a morphism $\alpha \in G\mapsto f_\alpha \in S(E)$ ($S(E)$ ...
3
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1answer
57 views

A non-singular quotient of $\mathbb{A}^n$ by a cyclic group is isomorphic to $\mathbb{A}^n$

Let $G$ be a cyclic group acting linearly on $X := \mathbb{A}^n$. If we assume that the quotient $Y:=X/G$ is non-singular, does it follow that $Y \simeq \mathbb{A}^n$? If so, is it necessary to assume ...
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2answers
110 views

Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
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2answers
26 views

Group acting on $X$ and element of normal subgroup $H$ fixes an element of $X$ implies $H$ fixes all of $X$

A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in ...
1
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1answer
45 views

Tensor power modulo cyclic group action

Let $M$ be some $R$-module and $n \geq 1$ be some positive integer. The cyclic group $\mathbb{Z}/n\mathbb{Z}$, with a chosen generator $t$, acts on $M^{\otimes n}$ via $t(m_1 \otimes \dotsc \otimes ...
0
votes
1answer
30 views

Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
2
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0answers
24 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 ...
1
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1answer
20 views

Dihedral Group question involving $D_{4n}$

Let $D_{4n}$ be the dihedral group of the order $4n$. Prove that $D_{4n}/\langle T^n \rangle$ is isomorphic to $D_{2n}$. We tried to configure an action on the diagonals of the $n$-gon and prove ...
5
votes
2answers
76 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
1
vote
1answer
38 views

Are there different combinatorial species with the same symmetry type?

First off: for my purposes, let $\sf B$ be the category of finite sets with bijections, and ${\sf B}_n$ the subcategory of sets with cardinality $n$, and define a combinatorial species to be a functor ...
1
vote
1answer
93 views

Calculate the homology group of $S^3/G$, an Harvard qualifying exam problem with “unclear” solution

Problem Suppose that $G$ is a finite group whose abelianization is trivial. Suppose also that $G$ acts freely on $S^3$. Compute the homology groups (with integer coeffcients) of the orbit space ...
1
vote
1answer
57 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
3
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2answers
45 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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1answer
19 views

Is the orbit map for a group action closed in this case?

Suppose a compact Lie group $G$ acts on a manifold $M$ and let $\pi : M \rightarrow M/G$ be the orbit map. Can I say that $\pi$ is closed map? If $C \subseteq M$ is a closed set in $M$ then I only ...
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0answers
28 views

G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
3
votes
2answers
26 views

Group Actions: Verify a Bijective Correspondence

This is an old exam problem: Given an action of $G$ on $X$, we can define $\varphi: G \to S_X$ by the rule $\varphi(g) = \sigma_g$, where $\sigma_g$ is left multiplication by $g \in G$. Prove that ...
2
votes
2answers
36 views

Does transitive imply it's the entire symmetric group

Let $G$ denote a finite group and recall that $G$ acts transitively (on itself) if and only if for all $x,y \in G$ there is a $g \in G$ such that $gx = y$. I am wondering if transitive may imply that ...
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3answers
23 views

The Stabilizer of the coset for the action of G on $G/H$ by left multiplication.

Let $H$ be a subgroup of $G$. What is the stabilizer of the coset $aH$ for the action of $G$ on $X=G/H$ by left multiplication? So, I think I've done this one correctly: The Stabilizer is of the ...
0
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1answer
24 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
0
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0answers
17 views

Continuous action on tensor product

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple ...
2
votes
1answer
12 views

Group orbit computations

Can someone please verify these? Compute the orbit of a vertex of the square under the tautological action of $D_4$. Compute the orbit of a point $x \in G$ under the action of $G$ on itself by ...
2
votes
1answer
44 views

the largest prime number $p$ dividing $ \left|G\right| $ is also divide $ \left|X\right| $.

Let $ G $ be a finite simple group which acts on finite set $ X $ non-trivially. The goal is that the largest prime number $p$ dividing $ \left|G\right| $ is also divide $ \left|X\right| $. I know ...
3
votes
0answers
29 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 ...