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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22 views

Is there a standard notion of a group “biaction”?

For a group $G$ there is a natural notion of left action on subsets of $G$ given by $g \triangleright H := gH$. But simultaneously, there is a natural right action as well: $H \triangleleft g := Hg$. ...
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2answers
28 views

Is there a contractible space with a free circle action?

Question in title. Seems no to me (some vague intuition here about contracting orbits to a fixed point), but I can't prove it. I'd prefer to be wrong. (I'm curious because I am thinking about group ...
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1answer
19 views

If a group acts properly and coboundedly on a hyperbolic space, each finite subgroup has a (uniformly) bounded orbit.

I am trying to solve the following problem: "Let $G$ be a group acting properly, coboundedly and by isometries on a hyperbolic space $X$. Show that there is a constant $C$ such that any finite ...
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1answer
35 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
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1answer
48 views

self-homeomorphism of the circle

$|z|=1$ is the unit circle in the complex plane. Suppose $g$ is a self-homeomorphism of this circle of order $n$, $n \in \mathbb{N}$, and $g$ acts freely. Is that true that $g$ must be defined by $z ...
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9 views

number of orbits by action of $D_{12}$ on $\mathbb{Z}_{12}^k$

Let $X=\mathbb{Z}_{12}^k$ for $k\in \mathbb{N}$ and $G=D_{12}$. Define an action of $D_{12}$ on $X$ by setting rotations $r^n(p)=(p_1+n,\dotsc,p_k+n)$ where the coordinates are taken modulo $12$ and ...
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22 views

Actions of unipotent groups

If we have an connected unipotent algebraic group $G$ over $\mathbb{F}$ (the algebraic closure of a finite field of characteristic $p>0$), with an $\mathbb{F}_q$ structure (where $\mathbb{F}_q$ is ...
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1answer
20 views

Why'' half-orbits'' of minimal $\mathbb{Z}$- action on compact Hausdorff space are still dense?

We say an action of $\mathbb{Z}$ on a compact Housdorff space $X$ minimal if every orbit of the action is dense in $X$. We assume the action is free and $X$ has no isolated points. Then in this case,...
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1answer
57 views

How do we call a map $F$ such that $F(g\cdot p)=\varphi(g)\cdot F(p)$?

Let $G$ and $H$ be groups acting on sets $M$ and $N$. Suppose that there is a group homomorphism $\varphi:G\to H$ and a map $F:M\to N$ such that $$F(g\cdot p)=\varphi(g)\cdot F(p)$$ for all $p\in M$ ...
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1answer
41 views

Group of deck transformations acts properly discontinuously

Let $M$ be a connected (smooth Riemannian) manifold which admits a universal cover $\tilde{M}$. Let $\Gamma$ be the group of deck transformations on $\tilde{M}$. I want to show that $\Gamma$ acts ...
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46 views

Group action on cosets

I would like to solve the Problem 2.19 from A Course in Modern Mathematical Physics by Szekeres. The problem is part of the paragraph 2.6 Group action. The formulation is: Problem 2.19 If $H$ is ...
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20 views

Descending group actions to coverings

Let $X$ be a path-connected space with universal cover $\widetilde{X}$, let $Y$ be another covering of $X$ $$ \widetilde{X} \hspace{1cm} \\ \searrow \\ \downarrow\hspace{.5cm} Y\\ \hspace{.25cm}\...
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50 views

Proving Sylow theorems without using group actions

Most of the proofs of Sylow theorems involves groups actions in some way as below: Sylow theorems - wiki There is a thread for them here, too: Proofs of Sylow theorems. However, I would like to see ...
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0answers
12 views

Polar of orthogonal set invariant under group action

I just ask the following question: Set invariant under group action Furthermore, How to prove the green part Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324) Let $$O(n)=...
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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. ...
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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 ...
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1answer
23 views

Properly discontinuous group actions - Hausdorffness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
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1answer
32 views

The set of prime ideals, whose contraction is a fixed prime in the ring of invariants, is finite.

Let $A$ be a domain and $G$ a finite group of automorphisms of $A$. I define $$A^G=\{a\in A\mid\sigma(a)=a ,\forall\sigma\in G\}.$$ Furthermore let $S\subset A$ be multiplicatively closed such that $\...
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1answer
19 views

Group action on algebra over a field defined on generators

Suppose $G$ is a group and $A$ is a finitely generated algebra over a field $\mathbb{k}$. Let $X=\{x_1,...,x_n\}$ be a set of generators for $A$, and suppose $G$ acts on $X$. Is this enough to define ...
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1answer
29 views

The action of automorphisms on the Riemann sphere

If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without ...
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1answer
18 views

How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...
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2answers
20 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
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1answer
82 views

Categorical Quotient and group actions

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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3answers
22 views

Misunderstanding the definition of a cycle (cyclic permutation)

Given a set $E$, let $\mathfrak{S}_E$ be the group of permutations of $E$. Definition.$\ \ $ Let $E$ be a finite set, $\zeta$ a permutation of $E$, and $\overline{\zeta}$ the subgroup of $\...
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1answer
32 views

The projective space as a homogeneous space

I want to understand why the projective space $\mathbb RP^n$ is diffeomorophic to $SO(n+1)/O(n)$? and why we can write the latter as $O(n+1)/O(n)\times O(1)$?
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49 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 ...
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1answer
29 views

Showing that $x$ is an element of group $G$ by left multiplication

$G$ is a group and $H \leq G$ with $|G:H|=3$. Show that $x$ is an element of $H$ if $x \in G$ with $|x|=7$. Hint: let $\langle x \rangle$ act on $G/H$ by left multiplication and look at the orbits. ...
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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 $G$...
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3answers
83 views

An erroneous application of the Counting Theorem to a regular hexagon?

I'm trying to count the unique orbits of a regular hexagon such that each vertex is either Black or White and each edge is either Red, Gree, or Blue. The group I've chosen to act on the hexagon is the ...
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1answer
37 views

In transitive (non-trivial) group action, there must be at least one group element without fixed point

Let a finite group $G$ act transitively on a finite set $S$ with $|S| \geq 2$. The problem is to show that not every $g \in G$ can have a fixed point in this action. I proved this on my own, but I'm ...
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1answer
27 views

Interpretation of the join of two stabilizer subgroups

Let $G$ be the group acting on two sets $X, Y$. Let $G_x$ and $G_y$ be stabilizer subgroups of some elements $x \in X, y \in Y$. It is easy to see that $G_x \cap G_y = G_{(x,y)}$, when we combine two ...
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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 $X=\bigcup_{i=1}^n G\cdot x_i$ ...
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1answer
58 views

Categorical Quotients and Group Actions on Varities

So I am given that Let $G = Z/dZ$ where d ≥ 1. Let w be a generator for G and let G act on $A^ {n+1}$ via $w(x_{0}, . . . , x_{n})$ = $(wx_{0}, . . . , wx_{n})$. How can I Show that the ...
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0answers
28 views

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$?

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$? This question is inspired by noting that if we have a Hamiltonian Lie group action $G\curvearrowright (M,\...
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1answer
46 views

If a subgroup has smallest prime index, then it is normal

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...
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2answers
52 views

How is the kernel of a group action defined?

Question: Show that the kernel of the group action of $G$ acting on set $A$ is equal to the kernel of the corresponding permutation representation of this action. I'm lost in this definition as ...
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1answer
42 views

Finite groups and prime divisors. Understanding how to deduce a claim from a certain proof.

In my algebra textbook, it goes like this. First, there is presented Cauchy's theorem: Let $G$ be a finite group, and let $p$ be a prime divisor of $|G|$. Then $G$ contains an element of order $p$....
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40 views

What does “factoring out an (group) action $\tau$ of a group $G$ acting on some set $E$” mean?

I am reading a survey article where they define the following objects: $\Gamma:=\mathbb{Z}^{n}$ seen as a group of translations. $\mathbb{T}:=\mathbb{R}^{n}/\Gamma$ is the $n$-dimensional ...
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1answer
19 views

Is it true that if all $G^\circ$ - orbits are closed in $X$ then all $G$ - orbits are closed in $X$?

Let $G$ be a Lie group acting continuously on a topological space $X$. Let $G^\circ$ be the connected component of the identity element of $G$ and let $[G:G^\circ]$ be finite. Then is the following ...
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58 views

G acts on X transitively, then there exists some element that does not have any fixed points

Let $X$ be a transitive $G$-set. ($G$ acts on $X$ transitively.) If $X$ is finite and has at least two elements, show that there is some element $g$ $\in$ G which does not have any fixed points; that ...
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1answer
39 views

The number of different G-actions on X [closed]

Let $X$ $=$ $\{$$1$, $2$, $3$$\}$ and $G$ $=$ $\mathbb Z_2$. How many different G-actions are there on $X$? Just learned group action. Need some hint on this one. Thanks.
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1answer
31 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
2
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1answer
58 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
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2answers
53 views

If $\sigma \in S_n$ has order some prime $p$, then is $|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p$? [closed]

Let $\sigma \in S_n$ be such that $o(\sigma)=p$ (some prime). Then is it true that $$|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p\ ?$$
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24 views

Every orbit $G\cdot x$ nonmeager is Baire.

For proof of Effros Theorem I have that $G$ is a Polish group and $X$ is a $G-$space Polish, but I need to show that if the orbit $G\cdot x$ is nonmeager then $G\cdot x$ is Baire in its relative ...
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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 ...
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1answer
99 views

Is $(\mathbb R^3\setminus \{0\})/\mathbb R^*$ a smooth manifold?

Let $G=\mathbb R^*$ act on $X=\mathbb R^3\setminus\{0\}$ by pointwise multiplication. That is for any $t\in\mathbb G$ and $(x_1,x_2,x_3)\in X$ we have $$t\cdot(x_1,x_2,x_3)=(tx_1,tx_2,tx_3)$$ Is ...
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2answers
34 views

Show that giving a right-action of a group $G$ on a set $A$ is the same as giving a left-action of $G^{op}$ on A

This is a part of an exercise from "Algebra: Chapter 0" by Paolo Aluffi. First, I provide the necessary definitions. An action of a group $G$ on a set $A$ is a set-function $\rho: G \times A ...
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1answer
30 views

References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
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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 ...