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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2answers
67 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
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0answers
17 views

Combinatorial proof of Rothe-Hagen

Wikipediate states the Rothe-Hagen identity below generalizes Vandermonde convolution: ...
2
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1answer
34 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 $2$. ...
2
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2answers
68 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
39 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 ...
2
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1answer
32 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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0answers
30 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
48 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, ...
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0answers
15 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
50 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
45 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
33 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 ...
12
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1answer
128 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 ...
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2answers
96 views

Orbit and Stabilizer

Are the following definitions essentially the same: Orbit: Let $G$ be a group of permutations of a set $S$. For each $s \in S$, let $\operatorname{orb}_G(s)= \{f(s) \mid f \in G\}$. The set ...
3
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2answers
40 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 ...
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1answer
43 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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1answer
37 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 ...
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1answer
45 views

Orbits of action of $SL_2(\mathbb{Z})$ on lattice

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$. Specifically, what are the orbits of this action?
3
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0answers
60 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 ...
3
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1answer
40 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. ...
5
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4answers
92 views

On what sets can $\mathfrak{S}_n$ act transitively?

I would like to know $\mathfrak{S}_n$ could act faithfully transitively on sets with $m$ elements, with $m > n$. I know that it is not possible if $m = n+1$ except for $n = 5$. Any ideas ?
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1answer
58 views

Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
3
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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 $fh<gh$ ...
5
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1answer
61 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 ...
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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 ...
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1answer
32 views

the group acts faithfully on the line

Let $G$ be a group. $G$ acts faithfully on the line $\mathbb{R}$ by orientation preserving homeomorphism, then does it imply $G$ is left ordered, i.e. there is an order $<$ on $G$, and if $a<b$, ...
0
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1answer
47 views

The action of free group on line

Let $G$ be a free group, if the action of $G$ on $\mathbb{R}$ is free, does it imply that $G$ is abelian?
3
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3answers
167 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
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2answers
34 views

Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a ...
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1answer
32 views

The function $A:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ given by $(n,x)\to nx$ is a group action on $\mathbb{R}$.

I somehow came through this True or False question, but before answering it, I just wanted to ensure some things. I understand that the axioms for $group-action$ must be met. Basically: ...
2
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1answer
33 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
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0answers
43 views

How many orbits are there in the group action of $S_4$ on $X={1,2,3,4}$?

So far I know that the order of the group is 4! which is 24 elements. I know that X the set has 4 elements. Finally, I understand that the Orbit-Stabilizer theorem states that: ...
1
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1answer
70 views

Transitive action of $\text{SL}(n,\mathbb Z)$

$\text{SL}(n,\mathbb Z)$ acts transitively on the set of ordered pairs of distinct 1-dimensional subspaces of $\mathbb Q^n$. Could you mention an article or a book where such a proof can be found? ...
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0answers
30 views

Standard action of SU(3) on $\mathbb{C}^3$

It is my understanding that SU(3) acts on $\mathbb{C}^3$ the same way SO(3) acts on $\mathbb{R}^3$, i.e. as proper rotations. If this is the case, then the orbit of $(1,0,0) \in \mathbb{C}^3$ ...
2
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0answers
20 views

Direct sums, tensor products etc. of $G$-vector bundles are again $G$-spaces

Given two $G$-vector bundles $E$ and $F$ over a $G$-space $X$ ($G$ some finite group), I am interested in the vector bundles $E \oplus F$, $E \otimes F$, $\operatorname{Hom}(E,F)$ etc. I am familiar ...
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0answers
62 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
0
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1answer
21 views

Action of the Unitary Group

I am working on the space $V_k (\mathbb{C}^n) = \left\lbrace (v_1, \cdots , v_k ) \in (\mathbb{C}^n)^k | \langle v_i, v_j \rangle = \delta_{ij} \right\rbrace $. I define the continuous action of ...
2
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1answer
41 views

Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
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0answers
29 views

Connected component and group action

Let $G$ be a topological group acting on a set $X$. Let $x \in X$ and consider the orbit $G.x$ endowed with the topology coming from the quotient $G/ Stab(x)$. If $G^0$ is the connected component of ...
2
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0answers
41 views

Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
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1answer
84 views

How can I use Clebsch-Gordan coefficients to decompose this group representation?

Let $G$ be a compact group, $\alpha$ be a unitary irrep of $G$ with carrier space $\mathcal A$, and $\beta$ be a unitary irrep of $G$ with carrier space $\mathcal B$. Then, the action of $G$ on ...
2
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0answers
34 views

Topologies of flag manifolds

I'm currently reading an article discussing flag manifolds and the action of $\mathrm{PSL}(n,\mathbb{C})$ on them. A flag (in my view at least) is a nested sequence $(y^1,\ldots,y^{n-1})$ of subspaces ...
1
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1answer
26 views

Intersection of stabilisers

I have a short question: if $G$ acts on $X$, is the intersection of all the stabilisers the same as the conjugates of the stabilisers? In other words does, for any $z\in X:$ $$\bigcap_{g\in G} ...
2
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1answer
20 views

transitively action of stabilizer of G

if $G$ acts transitively on $X$ and for a special $x$ in set $X$ we have the stabilizer of $x$ acts transitively on $X-\{x\}$ can we conclude that this proposition is true for all element of $X$?
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2answers
39 views

Prove this wreath product is a group [Homework]

I'm not usually one to post unworked problems here... I usually try to at least have an attempt, but unfortunately in this case I'm unable to even get an intuitive sense of what's going on here - and ...
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0answers
39 views

Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
3
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1answer
85 views

Why it is a group action?

Let a group $G$ acts on a vector space $V$ and let $f$ be a function on $V$. The action of an element $g \in G$ defined by the rule $g f(x)=f(g^{-1} x), \forall x \in V.$ A typical proof from a ...
2
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1answer
38 views

How many orbits are there in the group action $A\colon 3\mathbb{Z} \to\mathbb{Z}_{6}$ with action given by $(3n,m)=(3n+m)\bmod6$.

I am having difficulty trying to understand this question. All I know is that $3\mathbb{Z}$ and $\mathbb{Z}_6$ are both groups, that is: $$3\mathbb{Z}=\{\dotsc, -6, -3, 0, 3, 6 \dotsc\}$$ ...
1
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1answer
12 views

verifying the sizes of a group acting on a set

Is $S_4$ a group of size four and $X={1,2,3,4}$ a set of size four? Need clarification on this. just wan to be clear on the sizes of the group and the set.
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0answers
53 views

Group action problem.

I've formed up a group $G=\{(1),(34),(12)(34),(124)\}$ acting on a set $X= \{1,2,3,4\}$. Knowing the axioms for a group action, that is: (Compatibility with identity): $e*x=x$ for all $x\in X$ ...