Skip to main content

Questions tagged [group-actions]

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 $2n$ act 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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
10 views

Small examples of non-transitive Automorphism groups of Steiner Systems

I'm currently doing research for a bachelor's seminar talk. I have found a result from E. Mendelsohn, "On the groups of automorphisms of Steiner triple and quadruple systems" stating that ...
dilemmma's user avatar
2 votes
0 answers
27 views

An action of $\Gamma_0(N)$ having finitely many orbits

For positive integers $m$ and $N$ let \begin{align} M_{2, m, N}(\mathbf{Z}) = \bigg\{\gamma \in M_2(\mathbf{Z}) \; \bigg\vert \; \det(\gamma) = m, \gamma \equiv \bigg(\begin{matrix} \ast & \ast \\ ...
Joseph Harrison's user avatar
-2 votes
0 answers
28 views

covering space action implying a covering map [closed]

I am tried to prove the following claim: Let $X\circlearrowleft G$ be a covering space action, $H<G$ a subgroup. Then the projection $p:X/H\rightarrow X/G$ is a covering map. $x\underset{G}{\sim} ...
mappingmoe's user avatar
8 votes
1 answer
168 views

Can we view higher homotopy groups as symmetries?

If we want to think of a group as representing symmetry, we want it to act on something. In the case of the fundamental group of a (nice) topological space $X$, even though it's definition doesn't a ...
Chris's user avatar
  • 466
1 vote
1 answer
52 views

Riemannian Manifolds (Lee) Exercise 6-28 (d): Isometries converging pointwise converge topologically

Suppose $M$ is a connected, complete Riemannian manifold, $\mathrm{Iso}(M)$ is a smooth Lie group composed of all isometries of $M$, and $\phi_n$ is a sequence of isometries converging to an isometry $...
subrosar's user avatar
  • 4,794
0 votes
0 answers
62 views

The cohomogeneity one action of $\text{SO}(3)$ on $\mathbb{CP}^2$.

There is a cohomogeneity one action of $\text{SO}(3)$ on $\mathbb{CP}^2$ given as follows (thank you to @Thomas for the correction). There is an $\mathbb{R}$-linear action of $\text{SO}(3)$ on $\...
ABBC's user avatar
  • 101
0 votes
0 answers
40 views

Questions on a proof on $p$-constrained groups

Theorem: Let $G$ be a group and $p \in \pi(G)$. Furthermore, suppose that \begin{equation}\label{eq_p-constrained} C_{G/O_{p'}(G)}(O_p(G/{O_{p'}(G)})) \leq O_p(G/{O_{p'}(G)}). \end{equation} If $P$ ...
Stippinator's user avatar
0 votes
1 answer
86 views

$A_{11}$ has no subgroups of order $\frac{11!}{14}$?

This is a question posed on a qualifying exam from the past. I have elected to answer it. Let $A_{11}$ be the 11th alternating group (that is, $|A_{11}| = 11!/2$.) Prove that $A_{11}$ has no subgroups ...
roblich mandervach's user avatar
0 votes
0 answers
35 views

every orbit of this action has the same number of elements.

Let $G$ be a finite group acting transitively on a set $ X $, i.e., for every $x, y \in X $, there exists $g \in G$ such that $ gx = y $. Let $H$ be a normal subgroup of $G$. Consider the action of $H$...
Alex Nguyen's user avatar
-1 votes
1 answer
73 views

Is the quotient topology space with the quotient topology homeomorphic to the topological space?

Let $G$ be a group and $X$ a set. Then an action of $G$ on $X$ is a function $F : G \times X \to X$, where we write $F(g, x) = g \cdot x$, satisfying: For all $g_1, g_2 \in G$ and $x \in X, g_1 \cdot ...
Alex Nguyen's user avatar
4 votes
1 answer
135 views

Action of symmetric group on polynomial ring

In Example 1.1 of Eisenbud's Commutative Algebra he writes that the symmetric group $\Sigma = S_r$ acts on the polynomial ring $S = k[x_1, \dots, x_r]$ by $$\sigma(f)(x_1, \dots, x_r) = f(x_{\sigma^{-...
Zufallskonstante's user avatar
0 votes
0 answers
46 views

Explicit example of a set of coset representatives of $U(n)$ within $O(2n)$

I understand how to identify a unitary group $U(n)$ with the elements of the orthogonal group $O(2n)$ which commute with a linear complex structure $J$. I am also aware of the "two-out-of-three&...
Andrius Kulikauskas's user avatar
1 vote
0 answers
36 views

Proof of Thompsons $A \times B$-lemma

(Auxiliary lemma) Let $G$ be a $\pi$-group and $a$ a $\pi'$-element acting on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = 1$. Hey guys, I am having a ...
Stippinator's user avatar
2 votes
0 answers
44 views

How is the $\pi_1(X)$ action on higher homotopy groups visible in $\Pi_\infty(X)$

It is well known that $\pi_1(X)$ acts on all the higher homotopy groups, and this action can be seen in several different ways see this question. I have recently started working with the fundamental ...
DevVorb's user avatar
  • 1,495
1 vote
1 answer
50 views

Counting the number of rectangles whose vertices are among those of a regular polygon

Let $P$ be a regular polygon with $n$ sides inscribed in a circle. Suppose there exists a rectangle, all of whose vertices are among the vertices of $P$. Then show that $n$ must be an even number. ...
Ricci Ten's user avatar
  • 520
0 votes
1 answer
41 views

Example for application of theorem: $G = \langle C_G(a) \mid a \in Q \setminus \{1\}\rangle$

Let $p, q$ be distinct prime numbers, $G$ a $p$-group, and $Q$ a non-cyclic abelian $q$-group of automorphisms of $G$. Then, $$ G = \langle C_G(a) \mid a \in Q \setminus \{1\} \rangle . $$ Hey guys, ...
Stippinator's user avatar
0 votes
1 answer
27 views

Question regarding making the subsets of a set G a G-set by left translation

I am currently going through Bergman's notes for Lang algebra and came across this statement: If $G$ is a finite group and $x$ any subset of $G$, then the isotropy subgroup $G_x$ of $x$ under left ...
baslerbuenzli's user avatar
0 votes
0 answers
92 views

A weaker property than $2$-transitivity

Let us define a half $2$-transitive action of a group $G$ on a set $X$ as an action with the two following properties : 1° the operation is transitive; 2° for every distinct elements $x, y$ in $X$, ...
Panurge's user avatar
  • 1,827
0 votes
0 answers
13 views

2-homogenous actions

Let us define a half $2$-transitive action of a group $G$ on a set $X$ as an action with the two following properties : 1° the operation is transitive; 2° for every distinct elements $x, y$ in $X$, ...
Panurge's user avatar
  • 1,827
2 votes
0 answers
42 views

What action on $\mathbb{R}^2$ yields a closed genus 3 surface

The closed, connected, orientable surface of genus 3 has universal cover $\mathbb{R}^2 \rightarrow \Sigma_3$. As such, $\Sigma_3$ can be described as a quotient of $\mathbb{R}^2$ by some properly ...
JMM's user avatar
  • 1,165
0 votes
0 answers
23 views

Induced action on vector valued differential forms

Let $F(X)$ be the frame bundle of a smooth $n$-dimensional manifold $X$. Let $G$ be a finite subgroup of $GL_n(\mathbb C)$ acting by automorphisms on $X$ and let $V$ be a $GL_n(\mathbb C)$-...
Flavius Aetius's user avatar
2 votes
3 answers
120 views

A $G$-action on $X\times Y$ induces actions on $X$ and $Y$?

It's easily seen that the converse is true (simply define $g(x, y) := (gx, gy)$). However, while analyzing the original statement, I got stuck and soon realised that there is no reason for that to be ...
Atom's user avatar
  • 4,119
0 votes
1 answer
60 views

Number of group actions

Suppose that we have an action of a group $G$ on a $\Bbb{Z}$. This action corresponds to a homomorphism $\varphi: G \to Aut(\Bbb{Z}) \simeq C_2$. But we can have many homomorphisms of $G$ on $C_2$. Is ...
Greg's user avatar
  • 422
2 votes
0 answers
52 views

A question about the transitive subgroups of $S_4$

I'm trying to show that any transitive subgroup $H$ of $S_4$ is conjugated to one of the following subgroups: $V_4=\langle (12)(34),(13)(24)\rangle$, $C_4=\langle (1234)\rangle$, $D_4$, $A_4$ or $S_4$ ...
Simplicio's user avatar
0 votes
0 answers
8 views

When is the leaf of a foliation on the quotient by a properly discontinuous free group action isomorphic to a leaf on the total space?

Let $\tilde M$ be a manifold and $G$ a discrete group acting freely and properly discontinuously on $\tilde M$. Let $M:= M/G$ and $p: \tilde M \rightarrow M$ is the projection; then $p$ is a covering ...
rosecabbage's user avatar
  • 1,697
0 votes
1 answer
43 views

The number of orbits of the action of $G$ on $X$ is equal to $\frac{1}{|G|} \sum_{g \in G} |F(g)|$. [duplicate]

Proposition: The number of orbits of the action of $G$ on $X$ is equal to $\frac{1}{|G|} \sum_{g \in G} |F(g)|$. Proof: We consider a table $G \times X$, where we place a check mark if $g(x) = x$. ...
user avatar
1 vote
0 answers
28 views

Normalizer of an $A$-invariant Sylow $p$-subgroup

I was reading the Antionio Beltrán and Changguo Shao article On the number of invariant Sylow subgroups under coprime action and there is a part of the Lemma 2.5. which I do not undertand. First of ...
math_survivor's user avatar
1 vote
1 answer
141 views

Stability of stationary points

In the article by Hirsch "On stability of stationary points of transformation groups It's mentioned that $0$ is a stable stationary point of the diffeomorphism $f(x)=x+x^3$ (stationary point of ...
user56980's user avatar
  • 229
0 votes
1 answer
22 views

Counting orbitas for infinite group acting on an infinite set

If an infinite group $G$ acts on an infinite set $X$ with finite stabilizers for every element, then how many orbits could this action have? It seems that smaller stabilizers implies bigger orbits and,...
Greg's user avatar
  • 422
2 votes
0 answers
37 views

Tiling of a tree to show that a group acting freely on a tree is free

Let me start giving some context: Let $G$ be a group acting freely on a tree $T$. Let $T'$ be the barycentric subdivision of $T$ (that is, the graph obtained by placing a new vertex at the center of ...
ABC's user avatar
  • 904
0 votes
1 answer
41 views

How to think of regular orbits of a finite orthogonal group?

Let $G$ be a finite subgroup of $O(n)$ acting on $\mathbb R^n$. A regular orbit of the action is one such that the cardinality of the orbit is equal to $|G|$. I am at a loss as to how to prove some ...
rosecabbage's user avatar
  • 1,697
2 votes
1 answer
31 views

two-way monoidal orbit

Let $M$ be a monoid acting on a set $X$. We can define an equivalence relation on $X$ by $x \sim y$ iff $y \in M x$ and $x \in M y$. Given an element $x \in X$ we can let $R_x \subseteq X$ be the ...
Ben's user avatar
  • 579
2 votes
2 answers
141 views

Trouble with understanding classifying groups with semi direct products

I'm trying to understand the following strategy on classifying groups of a particular order from Dummit & Foote's Abstract Algebra (p.181): Let $G$ be a group of order $n$. You find proper ...
Ryan Zhou's user avatar
  • 110
1 vote
1 answer
46 views

Hamiltonian Group Actions on Calabi-Yau Cones

Let $(M, g, J, \omega, \Omega)$ be a Calabi-Yau cone (where $\Omega \in \Gamma(K_M)$ is the parallel holomorphic volume form), and assume we have a Hamiltonian group action $G \circlearrowright M$ ...
Albert Wood's user avatar
0 votes
1 answer
119 views

Defining group action formula for cartesian products

I am trying to understand @Martin Brandenburg's this proof. He said that "The group $H \times K$ acts on the set $HK \subseteq G$ via $(h,k) x := hxk^{-1}$" However I could not understand ...
user avatar
0 votes
0 answers
38 views

Group which acts properly on a tree is closed in the automorphisms group

Let $G$ be a locally compact group which acts properly on a locally finite (simplicial) tree $T$ (i.e., for each compact subset $K \subseteq T$ it holds that the set $G_K=\{g \in G| gK \cap K \neq \...
Bargabbiati's user avatar
  • 2,271
0 votes
1 answer
26 views

Existence of fundamental domain

I am trying to prove the existence of a fundamental domain for the free and properly discontinuous action of the group of deck transformations $\Gamma = \mathrm{Deck}(X)$ of the universal cover $\pi : ...
fresh's user avatar
  • 343
0 votes
1 answer
52 views

A question about smooth and effective finite group action

I encounter this question when trying to prove that if $f:O_{1}\to O_{2}$ is a diffeomorphism between two orbifolds $O_{1}$ and $O_{2}$, then the local group $G_{p}$ is isomorphic to the local group $...
Saiba Midori's user avatar
0 votes
1 answer
38 views

Construct covering space with given fiber

Suppose $(X, x_0)$ has a universal cover and $A$ is a left $\pi$-set. How do I find a covering map $q: Y \to X$ with the fiber of $x_0$ isomorphic to $A$ as a $\pi$ set? I vaguely know that this has ...
Nancium's user avatar
  • 383
0 votes
0 answers
30 views

Weyl group of $D_n$

We have that the root system of the Dynkin diagram of type $D_n$ can be realized by considering the standard inner product in $\mathbb{R}^n$ and the vectors $\{e_1 - e_2, \ldots, e_{n-1} - e_n, e_{n-1}...
Andreadel1988's user avatar
0 votes
1 answer
42 views

Matrices invariant under rotations are always proportional to the identity?

Is this proof true? Suppose we have a $3\times 3$ matrix $M^{ab}$ satisfying $$M^{ab}=R^a\,_cR^b\,_dM^{cd},$$ i.e. $$M=RMR^T,$$ for all rotations $R\in \mathrm{O}(3)$. Now, if denote representations ...
Ivan Burbano's user avatar
  • 1,268
1 vote
0 answers
32 views

Induced action on fundamental group

I encountered this problem on a practice exam (without sample solution): Identify $S^2$ with $\mathbb{C} \cup \infty$. Let $X_n = S^2\backslash\mathbb{Z}/n\mathbb{Z}$ denote $S^2$ minus the N-th roots ...
Nancium's user avatar
  • 383
-1 votes
1 answer
62 views

Alternative definitions outer semidirect product [closed]

I am working with outer semidirect products and have come across two definitions that are slightly different. Let $N,H$ be two groups and $\theta: H \to \text{Aut}(N)$ a group morphism. Then in both ...
noparadise's user avatar
1 vote
0 answers
14 views

Do amenable groups have invariant monotiles?

Let $G$ be a countable, discrete, amenable group, and let $B$ a finite subset. We say that $B$ is a monotile if $G$ is a disjoint union of translated copies of $B$. In "Monotilable Amenable ...
Susana Santoyo's user avatar
2 votes
0 answers
67 views

What is the name of the group of all circular slice rotations on the sphere?

Consider the following group that naturally acts on the $2$-sphere. An individual generator works by selecting a circle $c$ on the surface of the $2$-sphere, and then rotating $c$ in the plane ...
Sidharth Ghoshal's user avatar
1 vote
1 answer
45 views

"Equivariance" (?) of Adjoint action on $\mathfrak{g}$ and a $G \times V \rightarrow V$ action

I know the following theorem to be true (at least in some cases) but I don't know what this property is called or why it works. Let $G$ be a Lie group acting on a vector space $V$. This induces a $\...
Theo Diamantakis's user avatar
0 votes
1 answer
74 views

Transitivity Theorems - Can You Give Me Some?

Motivation. I get a question, and it reads as follows: "Assume $G$ is an abelian, transitive subgroup of $S_n$" (c.f. Dummit and Foote 4.1.3). Immediately, I can tell you a ton of properties,...
JAG131's user avatar
  • 917
4 votes
1 answer
108 views

Is the action of $\operatorname{Out}(G)$ on $[\operatorname{Rep}(G)]$ faithful?

Let $G$ be a finite group and let $\phi$ be an automorphism of $G$. We define an action of $\operatorname{Aut}(G)$ on the set $\operatorname{Rep}(G)$ of complex-valued representations of $G$ by ${}^\...
gimothytowers's user avatar
3 votes
1 answer
99 views

Prove $|M/G|=\sum_{x \in M} {\frac{1}{|G(x)|}}$

Let $G$ be a finite group operating on a finite set $M$. $|M/G|$ is the number of orbits. I want to prove that $|M/G|=\sum_{x \in M} {\frac{1}{|G(x)|}}$ This question is in preparation of Burnside-...
Cake's user avatar
  • 189
0 votes
0 answers
47 views

If $|C(g)|>|G|/2$, then g$\in Z(G)$?

I'm having trouble solving an exercise on characters of finite groups and I don't know what to do as I don't even understand the idea behind it. The question goes as follows : Let G be a finite group ....
GGG's user avatar
  • 347

1
2 3 4 5
64