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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Consider group G acting on a set X

Consider group G acting on a set X Give examples of: a)The action that is transitive and faithful My Answer: Group G under addition acting on a set of integers Z b)The action that is transitive ...
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29 views

Is the saturation of Borel sets Borel?

Problem. Let $G\times X\rightarrow X$ be a continuous action of a Polish group on a Polish space. Let $A\subseteq X$ be Borel. Is the saturation $[A]_{G}:=G\cdot A$ a Borel set? One approach. The ...
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53 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
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43 views

Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
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37 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
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41 views

is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
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24 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
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145 views

What is a “natural group action”?

Eg. The symmetric group on S acts on S in a natural way, for all sets S. Thanks in advance!
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39 views

Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states: Let $G$ act ...
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1answer
54 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
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32 views

for which values of k is the action of $S_A$ on k-element subsets faithful

I am solving dummit and foote question and I saw this question in one of the chapters. So in order to be faithful we have to have the given action induce distinct permutation on the set, but the ...
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8 views

cocompact action + finite stabilizers => proper action?

Assume a disctere countable group G acts on a smooth manifold M by diffeomorfisms and (1) M/G is compact and Hausdorff (2) all stabilizers are finite How to prove that the action is proper?
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28 views

Orbit-Stabiliser Theorem Applied

"The group S6 acts on the group Z6 via σ([a]) = [σ(a)], for σ ∈ S6 and a∈{1,...,6}. A permutation that is also an isomorphism is called an automorphism. The set G of automorphisms of Z6 is a group. ...
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104 views

What is a group action, and how can we apply it to Sylow theory

I am studying Sylow theorems at the moment, more specifically trying to solve the following problem that I recently posted: Let G be a finite group which has exactly eight Sylow 7 subgroups. Show ...
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123 views

If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. Now this is my ...
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1answer
118 views

Defining action of an elementary abelian 2-group on a vector space.

I have a group $G = \oplus_{\alpha} \mathbb{Z}/2$, where all the direct summands are indexed by the elements of a set (a list in GAP). I want to define in GAP an action of this group on a vector ...
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1answer
38 views

Proving that the dihedral group $D_n$ has $2n$ elements

I am trying to prove that the dihedral group $D_n$ has $2n$ elements by using the theory of group actions. Specifically I want to use the orbit stabilizer theorem. So I need $D_n$ to act on a specific ...
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96 views

Reference for theorem on non-decreasing functions of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose $k\colon M\rightarrow N$ is a function such that $k(\epsilon)=\epsilon$ for all $a,b\in M$, there exists a unique $v\in N$ such ...
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2answers
55 views

Difference between Stabilizer and Centralizer?

I know that the Centralizer of an element $a$ in a Group $G$ is defined as follows $$C_G(a) = \{ g \in G \space | \space ga = ag \}$$. It can also be defined as follows $$C_G(a) = \{ g \in G \space ...
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17 views

What are the $SL(2, \mathbb{R})$-invariant Baire measures on $\mathbb{R}^2$?

This was an interesting problem that came up in qual studying. $SL(2, \mathbb{R})$ acts on $\mathbb{R}^2$, and hence on any measure on $\mathbb{R}^2$. What are the Baire measures (i.e. Borel ...
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277 views

Elementary Combinatorial Proofs using group action

In trying to prove that the number of spanning trees in $K_5$ is $125$ I adopted the following method: Let $S$ be the set of all such spanning trees and let $S_5$ act in a natural way on $S$. Now ...
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1answer
91 views

Show that every $A \in SL_3-$action has at least 3 fixed points on $\mathbb{P}^2$.

Consider the natural action of $SL_3(\mathbb{C})$ on $\mathbb{P}^2$ via: $$ SL_3(\mathbb{C}) \times \mathbb{P}^2 \to \mathbb{P}^2, \ (A,[v]\mapsto [Av]). $$ It is clear that the kernel of the ...
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69 views

Orbits of properly discontinuous actions

Definition Let $G$ be a group and $X$ a topological space. Let $G\curvearrowright X$ by homeomorphisms. We call the action properly discontinuous if for all $x\in X$ there exists an open neighborhood ...
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33 views

How can I obtain these differential operators for this transformation?

I have transformation as the following form \begin{eqnarray} \begin{split} &u \longrightarrow \bar{u}=(ax+by+\eta)^{-3} u,\\ &x \longrightarrow \bar{x}=\frac{\alpha x+\beta ...
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1answer
88 views

Is $BG =EG / G$ a CW complex?

am currently working through J.Rosenbergs construction of classifying Spaces for the +-construction of higher K-Theory. He defines a CW-complex EG as the direct limit of the inductively defined Spaces ...
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15 views

Orbit space of $S^5$ under certain action

Let $Z/6 = <w| w^6 = 1>$.Consider a $Z/6$-action on $S^5$ as generated by the relation $w.(z_1 , z_2 , z_3 ) = (wz_1 , w^2z_2 , w^3z_3)$ ,where $(z_1,z_2,z_3)$ belongs to $S^5$. Question 1: ...
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2answers
46 views

what is the role played by the group in group action?

$H = Z_5 $ and $G = S_5$ and let $A =\{1,2,3,4,5 \} .$ My question is 1.What is difference between the the action of H on A and G on A ? 2.How many group actions are possible in the above a cases ? ...
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1answer
65 views

ordinary cohomology from equvariant cohomology

Is it possible that the ordinary cohomology of a space can be obtained from its equivariant cohomology? action is algebraic torus action and space is nonsingular complete complex algebraic variety ...
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1answer
76 views

Prove: There is a $g \in G$ such that $\forall$ $x \in X: g \circ x \neq x$

I have to prove this theorem for my math study: Let $G$ be a finite group, and $X$ a set with #$X \geqslant 2$. Let the action of $G$ on $X$ be transitive. Prove: There is a $g \in G$ such that ...
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1answer
34 views

computing the orbits for a group action

Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then the number of orbits for the action of $G$ on the fields with nine elements is 3 5 6 9 I have ...
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1answer
41 views

Role of Group actions in Differential Geometry

This is a rather soft question, my hope is to bring some order into the stuff I would like to learn about differential geometry -- here it is: I was told over and over again that Geometry has to do ...
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35 views

Let a group G act on a set $X$, and suppose that $x,y \in X$ lie in the same orbit. Prove that $G_y=g^{-1}G_xg$ for some $g \in G$

Let a group G act on a set $X$, and suppose that $x,y \in X$ lie in the same orbit. Prove that $G_y=g^{-1}G_xg$ for some $g \in G$ Ok, lets assume $G$ acts on $X$,where $x,y \in X, g \in G$ and ...
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1answer
40 views

Explain why $g(Hx)=H(gx)$ is not an action of a group $G$ on the set of all left cosets of $H$ in $G$

Question: Explain why (xH)g=(xg)H is not an action of a group G on the set of all left cosets of H in G Im only just learning about acts on groups and need help understanding how to work this ...
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1answer
57 views

A finite group of order $pq$ cannot be simple.

Let $p$ and $q$ be prime numbers. I wish to prove that a finite group $G$ of order $pq$ cannot be simple. Proof. Case 1: $p\not= q$. Case 2: $p=q$. Consider the first case where $p\not= q$. Without ...
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1answer
37 views

Orbits of a 4x4 checkerboard

Suppose we have a $4\times4$ checkerboard with 16 squares. Let $S$ be the set of all colorings with 8 black and 8 white colors. Note: colorings are considered the same if they map to each other using ...
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1answer
44 views

Discriminant is the unique invariant of $\text{SL}_2\mathbb{Z}$ acting on polynomials.

The following is a really wonderful theorem that I really have no idea how to prove. Consider $p=ax^2+bxy+cy^2$, and let $\text{SL}_2\mathbb{Z}$ act on all such $p$ by $\begin{pmatrix} a&b \\ ...
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1answer
214 views

The sum of orbit size of some element over the image of group “polynomial”

$\DeclareMathOperator{\orb}{orb}$ Say I have a group "polynomial", $p$, on $S_n$, that is $p(x)=a_1 x^{\epsilon_1}...a_n x^{\epsilon_n}$ for all $x \in S_n$, fixed $a_i \in S_n$ and fixed $\epsilon_i ...
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2answers
30 views

If $X$ and $Y$ are g-equivariant homeomorphic then $X/G$ and $Y/G$ are homeomorphic

Let $X$ and $Y$ be $G$-sets (That is the group $G$ acts on $X$ and $Y$). We say that the function $f: X\to Y$ is G-equivariant if $f(g.x) = g.f(x)$ for all $x\in X$ and all $g\in G$. Prove that if ...
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2answers
40 views

Group action and Right action

Sorry if this may seem trivial - I just started studying Group Theory. This is the problem: Prove that $(g,h) \rightarrow hg$ does not define a group action with $g$ acting on $h$. Prove instead ...
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35 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
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38 views

Intuition behind group action on a set

In Algebra Chapter 0 the definition of a group action on a set is given as: An action of a group $G$ on a set $A$ is a set function $P:G\times A\rightarrow{A}$ such that $P(e_G,a)=a$ and ...
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3answers
57 views

How many faces, edges and vertices are fixed when $S_4$ permutes the diagonals of a cube?

Consider the action of $S_4$ on a cube, where it acts by permuting the long diagonals. The conjugacy classes of $S_4$ are denoted by $id$, (12), (123), (1234) and (12)(34). I want to know the number ...
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1answer
69 views

Computing the size of the stabilizers when $U(q)$ acts on $\Bbb Z_q$

Let $\mathbb{Z}_q$ be the additive group of integers modulo $q$ and $U(q):=\{g\in\mathbb{Z}_q:(g,q)=1\}$. If $a\in\mathbb{Z}_q$, then what is the cardinality of the set $\{g\in U(q):ga\equiv a(\mod ...
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1answer
19 views

Poisson actions defined in terms of coactions.

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, ...
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1answer
43 views

Action of automorphisms on Eisenstein series

Let $ f \in \mathcal{M}_{k}(\Gamma) $ and $ \sigma \in \textit{Aut}(\mathbb{C}) $. Suppose $$ f = \sum_{n=0}^{\infty}a_{n}q^{n} .$$ Then we define the action of $ \textit{Aut}(\mathbb{C}) $ on $ ...
3
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1answer
42 views

Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...
3
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1answer
77 views

Isomorphic but not equivalent actions of a group G

This is in some sense a continuation of this problem. Given a group $G$ I would like to exhibit two actions of $G$ on a set $[n] =\{1,\ldots,n\}$ such that the two actions are isomorphic yet not ...
3
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1answer
23 views

how to see whether a bundle is trivial or not?

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
3
votes
2answers
63 views

Group action with two orbits

Suppose a group $G$ acts faithfully on a set of five elements, inducing two orbits of size $3$ and $2$ respectively. What group may $G$ be? There is clearly a homomorphism $G \mapsto S_3$ and ...
0
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0answers
15 views

equivariant map and cocompact action on subgroup.

Could anybody help me with the following? Let $G,H$ be two hyperbolic groups acting on $X$ and $Y$ respectivly. Let $f:G\rightarrow H$ be map between these groups and let $q:X\rightarrow Y$ be a an ...