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 views

Homomorphism from a subgroup to a group is injective.

I'm reading a proof and I don't quite understand one step of the proof. We want to deduce that if G acts transitively on A then $ \bigcap_{\sigma \in G} \sigma G_{a} \sigma^{-1} = 1$. (Where $G_{a}$ ...
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0answers
11 views

Free action of symmetric group

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$? Is there a compact manifold which can be act freely by all symmetric ...
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0answers
10 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
4
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3answers
72 views

Involution on Cantor space with exactly one fixed point

Let $X=\{0,1\}^{\mathbb{N}}$ be the Cantor space. What is an example of a continuous map $\sigma : X \to X$ with $\sigma^2=\mathrm{id}$ and $\# \{x \in X : \sigma(x)=x\} = 1$? This has to exist, ...
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0answers
24 views

Help with terminology

I need some help unraveling the terms that appear in the following passage. I found it in a book on some conference proceedings related to Differential Geometry. Let $f:X \to R^3$ be a smooth curve ...
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0answers
25 views

Definition of Representation in terms of Group Action

The definition of a representation of a group $G$ over a vector space $V$ is a map $p: G \to GL(V)$. According to wikipedia, for finite groups an equivalent definition is an action of $G$ on $V$. ...
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1answer
32 views

Fundamental domains of Dihedral groups

Let $D_n$ be dihedral group of order $2n$, it acts on plane $\mathbb{R}^2$ in a standard way, by rotations and reflections. How one can find fundamental domains for such action?
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2answers
27 views

How do open subsets of X/G look like?

Let $G$ act continuously on $X$, where $X$ is a topological space. So I wonder about how open subsets look like in $X/G$. The action $a$ is defined as $a(g,x)=g.x$.
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0answers
19 views

Group action on a tensor product

Let $R \subset S$ be an extension of commutative rings, $G$ a group and $M$ a left $R[G]$- module. Then how do I make the tensor product $S\otimes_R M$ into a left $S[G]$- module? What is the action ...
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2answers
15 views

Definition of equivariant map if one of the actions is a right action

Let $G$ be a group and $f:X\to Y$ a map between two $G$-sets which preserve the $G$ action. If $X$ has a left $G$ action and $Y$ right $G$ action then why do we define $f(g.x)=f(x).g^{-1}$ for all $ ...
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1answer
28 views

Orbit spaces of linear involutions on spheres

Consider the following involutions ($\mathbb{Z}_2$-actions) on the unit $2$-sphere $S^2 \subseteq \mathbb{R}^3$: $(x, y, z) \mapsto (-x,-y,-z)$, the antipodal action; the orbit space is the ...
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1answer
43 views

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I've come up with a proof for the following question, and I'd like to know if it's correct (as I couldn't find anything online along the lines of what I did). Question Let $p$ and $q$ be primes ...
2
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0answers
41 views

Hausdorffness of quotient space

Let $G$ be a compact topological group, and $X$ be a Hausdorff space. We assume that $G$ acts on $X$. Is the quotient space $X/G$ with the quotient topology a Hausdorff space? It seems that the ...
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1answer
45 views

How to think of group actions?

I am a little confused on how exactly I should be thinking of an action on a group. I have been trying to read up on it and came across Timothy Gower's blog which I think does a good job explaining ...
1
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1answer
32 views

Group actions and permutation representation

Im trying to solve this problem from Dummit & Foote: Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma\in G$ ...
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1answer
32 views

Group action of a direct product of groups

Let $G$ be a finite group acting on the $n$-dim vector space $X$. Let $R$ be an $n$-dim representation of $G$. $X$ consists of points $(x1,x2,...,xn)$, which are acted upon by $R(g)$, for $g$ in ...
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1answer
32 views

Index of center $Z(G)$ is finite implies the number of elements of conjugacy class is finite

Exercise Let $G$ be a group such that its center $Z(G)$ has finite index in $G$. Show that every conjugation class has finite elements. I don't know how to attack the problem. I thought the ...
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0answers
36 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 ...
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1answer
39 views

Action of Symmetric Group on Lie Polynomials with GAP

Let $L$ be the free Lie Algebra, freely generated by $x_1,x_2, \ldots, x_n$. Let $f$ be a polynomial in $L$ and $\sigma \in S_n$, how to do $\sigma$ act on $f$ in GAP? That is $$\sigma f(x_1, \ldots, ...
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0answers
52 views

Why is this action faithful? (Proof that the automorphism group of $A_n$ is $S_n$ for $n\geq 7$.)

I'm currently trying to work through a proof that Aut$(A_n) \cong S_n$. In particular I'm looking at theorem 2.3 (on page 18) in R. Wilson's book 'The Finite Simple Groups'. (Click here for a download ...
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2answers
18 views

How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...
2
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0answers
24 views

Sufficiently transitive implies alternating sans Enormous Theorem

According to this webpage and this mathworld article, if $G<S_n$ is a permutation group which acts sextuply transitively then $G=A_n$ is the alternating group, but this fact is known on the basis ...
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1answer
29 views

Product of two stabilizers of transitive group action is proper subset of G?

Suppose $G$ is a finite group and G acts transitively on some set $X$. Let $a$ and $b$ be two distinct elements of $X$ and $G_{a}$ and $G_{b}$ be stabilizers of $a$ and $b$ respectively.Show that ...
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1answer
43 views

Orbit space of $S^n \times S^n$ under the antipodal action

Write $S^n$ for the $n$-dimensional sphere, the space of vectors of length $1$ in $(n+1)$-dimensional Euclidean space. Consider the antipodal action on $S^n$, i.e. the action of $\mathbb{Z}_2$ given ...
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2answers
158 views

Proving a group, $G$, is a group action onto some set, $X$

I want to prove that a function defines a group action: We have group $G$ of diagonal $2\times 2$ matrices under matrix multiplication, and the set $X$ of points of the Cartesian plane, eg: ...
2
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1answer
57 views

Number of actions of $\mathbb Z$

Let $X$ be a finite set. Determine the number of actions of $\mathbb Z$ on $X$. If $X$ is a finite set with $|X|=m$, then $|\{f:X \to X : \text{f is bijective}\}|=m!$. Finding the number of actions ...
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2answers
39 views

Normal subgroup acting on a set

I am trying to solve the following problem: Let $G$ be a group acting on a set $X$ and let $S \lhd G$. Determine the necessary and sufficient conditions so that there exists an action of $G/S$ on $X$ ...
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1answer
65 views

Confusion about Actions of the Symmetric Group

I'm working on some practice questions and I am having trouble understanding actions of the symmetric group. I have the answers, but there were no explanations as to how they were derived. I feel ...
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1answer
57 views

Prove a result on transitive group actions.

Let $G$ be a group and $A$ & $B$ be two sets s.t. $G$ acts transitively on each of $A$ & $B$. Choose some $\alpha$ and $\beta$ in $A$ & $B$ respectively then prove that if $G=G_\alpha ...
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2answers
39 views

Regular Group Actions and Functions

An action of a group $G$ on a set $X$ is regular if for any $x,x'\in X$, there exists a unique $g\in G$ with $gx=x'$. (a) Give an example. (b) Let $X$ be a set and let $G$ be a group which acts ...
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0answers
24 views

Combinatorial proof of Rothe-Hagen

Wikipediate states the Rothe-Hagen identity below generalizes Vandermonde convolution: ...
2
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2answers
76 views

For a transitive permutation group $G,$ show that there is some $\sigma \in G$ such that $\sigma(a) \neq a$ for all $a \in A.$

Here is a problem that I have been working on. I was able to prove part A, but am having problems with part B. Thanks! Let $G$ be a permutation group acting on a finite set $A.$ If $g\in G,$ let ...
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1answer
52 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
2
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1answer
48 views

Riemannian symmetric pair $(G,H)$ with H non-compact

Let $G$ denote a connected Lie group and $H$ a closed subgroup. Suppose that $\sigma$ is an involutive automorphism of $G$. Assume that $(G,H,\sigma)$ is a Riemannian symmetric pair. So far I have ...
2
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2answers
55 views

How to describe $G/U$?

Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, ...
2
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1answer
37 views

The Weyl group of $E_6$ acting on embedded circles

I want to know the number of components of the normalizer of an arbitrary circle subgroup $S$ of (the compact real form of) the exceptional Lie group $E_6$. This number will always be $1$ or $2$. ...
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0answers
21 views

Conjugate actions

What is the definition for two group actions to be conjugate? For example a smooth action of a finite group on a manifold is locally conjugate to an orthogonal action.
3
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4answers
51 views

Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
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0answers
47 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
0
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0answers
46 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
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1answer
40 views

relation between eigen values

Let $W$ be a finite subgroup of $GL(V)$ and hence it acts on $V$. Now consider the contra gradient action of $W$ on $V^*$. Now how to show that the eigen value of this action is the reciprocals of the ...
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1answer
129 views

Is $GL_2(\mathbb Z)\cdot X$ a dense subset of $\mathbb R^2$?

We know that the set $D=\{a+b\sqrt{2} \mid a,b\in \mathbb Z\}$ is dense in $\mathbb R$ because $D$ is a subgroup of $(\mathbb R,+)$ that is not of the form $\alpha \mathbb Z$. So, the following set ...
2
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1answer
49 views

Double cosets and conjugation

Let G be a group and $h,g \in G$ with $SgT=ShT$ Show that the subgroups $gTg^{-1}\cap S$ and $hTh^{-1}\cap S$ are conjugated in S. Two subgroups $U,V$ are conjugated if $\phi(U)=gUg^{-1}=V$, right? ...
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votes
1answer
40 views

Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
3
votes
2answers
44 views

Index and normal subgroups

I want to show the following. For an infinite group G with only two normal subgroups (G and {e}) holds: There does not exist a non-trivial subgroup of G with finite index. I think i should prove ...
3
votes
1answer
43 views

The index of the Core of a group

I have to prove the following: Let $G$ be a group and $U$ be a subgroup of $G$. Then it holds: If $U$ has finite index, then $\text{Core}_G(U):=\bigcap\limits_{g\in G}gUg^{-1}$ has also finite index. ...
3
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0answers
74 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 ...
5
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1answer
65 views

Recovering a group action from sizes of orbits of individual elements

Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...
1
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1answer
30 views

For $f \in \mathbb{R}[T_1,\dots,T_n]$ and $\sigma \in S_n$, the number of different polynomials of the form $\sigma(f)$ divides $n!$

I have the following problem that I believed I solved but am trying to understand better. Let $n\geq 2$. Consider $\sigma \in S_{n}$ and $f \in \mathbb{R}[T_1,\dots , T_n]$, let ...
3
votes
1answer
51 views

The group acts on an ordered set

Let $G$ be a group, $G$ acts on an ordered set and preserves its order, i.e. $a<b$, then $g(a)<g(b)$ for $g\in G$. Then does it imply there is a left order on $G$, i.e. $f<g$, then $fh<gh$ ...