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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Group Action and Orbits

I am looking at the following example which says find the orbit of $0$ under addition by $2$ and $3$ if $\mathbb{Z}_4$ acts on itself by addition. So to find the orbit of $0$ we are looking at the set ...
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1answer
16 views

Free and proper action

I don't know how to solve this problem. Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by ...
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2answers
56 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
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1answer
28 views

$AGL(V) = V \rtimes GL(V)$ with $GL(V)$ acting from the right

For a vector space $V$, I have constructed $AGL(V) = V \rtimes GL(V)$ as the elements $(v, A) \in V \times GL(V)$ (Cartesian product of sets, not a direct product) with multiplication $(v, A) (w, B) = ...
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0answers
15 views

Symmetric groups and transitive action

I am trying to show that for $(x,x_1),(y,y_1)$ there exists $g\in S_n$ such that $gx=y$ and $gx_1=y_1$ where $x$, $x_1$, $y$, $y_1\in \{1,2,3,\dots ,n\}$ and $x\neq x_1$, $y\neq y_1$. Is this claim ...
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1answer
42 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
3
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1answer
19 views

Transitive action on Poincare upper half plane

I am trying to prove that the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ via $$ \left( \begin{array}{ c c } a & b \\ c & d \end{array} \right)z\rightarrow \frac{az+b}{cz+d} $$ ...
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0answers
27 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
25 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 ...
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3answers
78 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
25 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
28 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
20 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
16 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
29 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
46 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 ...
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0answers
42 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
46 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 ...
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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
40 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 ...
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0answers
25 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 ...
4
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1answer
30 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 ...
5
votes
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: ...
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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
40 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
77 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 ...
3
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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)
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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 ...
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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
38 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.
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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 ...
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0answers
47 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 ...
12
votes
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
votes
1answer
51 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? ...