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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1answer
33 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 $a<b$, ...
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1answer
51 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?
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3answers
187 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
47 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
33 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: ...
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1answer
35 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
32 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 $(1,0,0) \in \mathbb{C}^3$ ...
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1answer
73 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? ...
2
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0answers
21 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
63 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 $E$ ...
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4answers
94 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
23 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
votes
2answers
51 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 ...
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1answer
96 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
39 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 ...
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1answer
36 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
votes
1answer
21 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
43 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
91 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 ...
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1answer
43 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: $$3\mathbb{Z}=\{\dotsc, -6, -3, 0, 3, 6 \dotsc\}$$ ...
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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
56 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 $x\in X$ ...
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0answers
37 views

Group action of $G<\mathbb Z^\infty_2$ over the Golden mean shift

I'm am looking for an action of an infinite subgroup of $\mathbb Z^\infty_2$ over the golden mean shift space $$X=\{x\in \{0,1\}^\mathbb N : x_i=1\Rightarrow x_{i+1}=0\}$$ such that any element of $G$ ...
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1answer
37 views

show that there is some element x∈X whose stabilizer Gx is all of G where G is a group of order p^k, where p is prime and k is a positive integer

I'm having trouble with this problem: Suppose that G is a group of order p^k, where p is prime and k is a positive integer.
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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: ...
0
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1answer
96 views

QUESTION about a regular dodecahedron [closed]

Can someone help me with this question? I have no clue. Thank you so much! Let G denote the group of rotational symmetries of a regular dodecahedron. This problem invloves considering the action of G ...
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3answers
90 views

If G acts on X, show that there must be a fixed point for this action. Please help. [closed]

Suppose that G is a group of order p^k, where p is prime and k is a positive integer. Suppose that X is a finite set and assume that p does not divide the size |X| of X. If G acts on X, show that ...
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1answer
40 views

If two groups act on a set in the same way then are the two groups related?

Let $G_1, G_2$ be two groups that act on a set $S$ on the left, such that for all $g \in G_1$ there's $g' \in G_2$ such that $g\cdot s = g'\cdot s$ for all $s \in S$. Define $h : G_1 \to G_2$, $h(g) ...
0
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0answers
44 views

Burnside's Lemma and Stirling Numbers of the First Kind

I've seen that $n!=\displaystyle\sum_{p=0}^n s(n, p)n^p$, where $s(n, p)$ are the signed Stirling Numbers of the First Kind, whose absolute values count the number of permutations in $S_n$ which have ...
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2answers
78 views

Action of $SO_n$ on $\mathbb{S}^{n-1}$ induces fibre bundle.

Real compact Lie group $SO_n$ acts smoothly and transitively on $\mathbb{S}^{n-1} \subseteq \mathbb{R}^n$ with obvious action. Isotropy subgroup of each point in $\mathbb{S}^{n-1}$ is isomoprhic to ...
2
votes
1answer
81 views

Representation of $GL_2$ on $K^2$

In one of my problems it says the following: Let $K$ be an infinite field. Consider the linear action of $GL_2$ on $K[x,y]$ induced by the natural representation of $GL_2$ on $K^2$. I don't know what ...
1
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1answer
21 views

Cardinality of rational exponentiation orbit space

Let $X=(0,\infty)$ be the set of positive real numbers. Let $G=\mathbb{Q}\backslash\{0\}$ be the multiplicative group of rational numbers. $G$ acts freely on $X$ by exponentiation: $r\cdot x=x^r$ for ...
1
vote
1answer
31 views

Why are isotropy groups named as such?

Why are isotropy groups, also known as stabilizers, named as such? In physics, the word isotropy means having the same property in all directions. Can one draw an analogy from this to interpret the ...
7
votes
1answer
59 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
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vote
1answer
63 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 $A^G$. ...
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0answers
53 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
2
votes
0answers
42 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
46 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?
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0answers
21 views

Circle action on the product of a Mobius band and a circle.

Consider the product of a Möbius band and a circle $Mo\times S^1$. Is there a circle action on $Mo\times S^1$ such that it is equivariantly homeomorphic to the twisted product $D^2 ...
5
votes
1answer
132 views

If a finite group $|G|$ acts transitively on a set $X$ with $|X|=2^n$, $n \geq 1$, then $G$ has an involution with no fixed points

Let $G$ be a finite group acting transitively on a set $X$, where $|X| = 2^n$ for some $n \geq 1$. Show that some element of $G$ acts as an involution with no fixed points. While it is fairly easy ...
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0answers
316 views

Can we prove $H \cong xHx^{-1}$ given $H \le G, x \in G$ using group action?

The exercise is as follows: $G$ is a group, $H \le G$. For any $x \in G$, to prove that $H \cong xHx^{-1}$. I am able to prove this isomorphism by defining a bijection $f : h \mapsto xhx^{-1}$ ...
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0answers
7 views

$\bar{L}$ points of $GL_{(ab)^2}/PGL_a\times PGL_b$

I am reading the paper "Matrix invariants of composite size" by A. Schofield (see here), and I have trouble understanding some one of his arguments, and I hope someone can explain them to me. He ...
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0answers
25 views

How to write Cayley representation permutations as cycles?

The representation of $g \rightarrow xg$ was given to us as Cayley representation. I believe it means that every group element $g \in G$ is mapped to another element $xg$ where $x$ is the same for ...
0
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1answer
23 views

Compute $S_3$ acting by conjugation on the set $X$ of $6$ subgroups of $S_3$

I know that the subgroups of $S_3$ are $\{e\}$, $\langle(12)\rangle$, $\langle(13)\rangle$, $\langle(23)\rangle$, $A_3$, and $S_3$. What I also know is that conjugation is $C_g(H) = gHg^{-1}$. Thus in ...
2
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0answers
29 views

Finding a fundamental polygon for two-generator subgroup of PSL(2,R)

Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like ...
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0answers
52 views

Action of a Lie group, a map of constant rank

Consider some Lie group $G$, smooth manifold $X$ and some action of $G$, i.e. a group homomorphism $\mathcal{A}: G\longrightarrow \mathrm{Diffeo}(X)$ such that the map $(g,x)\mapsto ...
0
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0answers
20 views

six transitive permutation groups

If I'm explaining right than please give me some hints about how we prove a permutation group is six transitive. I have proved that it is two transitive because stabilizer of one point acts ...
0
votes
1answer
34 views

Question about May's Algebraic Topology book

I am referring Google Books for the question: link in the proof of the first lemma, why is $hns=\phi(hs)$ true? I simply cannot get it...