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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48 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
47 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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0answers
37 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
63 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
41 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
47 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
221 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
33 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
43 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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1answer
38 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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2answers
48 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
67 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
20 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
44 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
83 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 ...
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1answer
24 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 ...
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2answers
69 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 ...
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0answers
17 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 ...
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0answers
25 views

Question on proper group actions

I am trying to prove the following - If $G$ is a discrete topological group acting properly on a Hausdorff space $X$ then the isotropy group (also called stabilizer) $G_x$ of $x \in X$ is finite. ...
3
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1answer
52 views

First examples for topology of non-Hausdorff spaces

I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions). As a first example, ...
2
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1answer
50 views

Permutation isomorphic subgroups of $S_n$ are conjugate

Consider $G,H \leq S_n$ and their natural action on $[n] = \{1,\ldots,n\}.$ We say that $G$ and $H$ are permutation isomorphic if there is a bijection $\varphi:[n] \mapsto [n]$ and group isomorphism ...
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1answer
35 views

Describe invariants using coaction.

Let $X$ be an algebraic variety and $G$ be an algebraic group which acts on $X$. We know that an invariant function $f$ in the coordinate ring $\mathbb{C}[X]$ is a function such that $g(f) = f$ for ...
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1answer
25 views

If $Aut_K(L)$ operates transitive over the zeros of $f$, then $f$ is irreducible

Let $K$ be a field, $f\in K[X]$ a separable polynomial and $L$ a splitting field of $f$. Show that if $Aut_K(L)$ operates transitive over the zeros of f, then f is irreducible. Can someone help?
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1answer
27 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
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1answer
67 views

Number of ways to pick N numbers from 0,1,…,N-1, with possible duplication, with sum equal 0 mod N

We have the numbers $0,1,2,....,N-1$ in $\mathbb Z_N.$ I want to pick $N$ numbers from these. These are the rules: Duplication may occur We don't care about ordering, $00041$ is equivalent to ...
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1answer
69 views

Group action on $\mathbb R^2$: are my thoughts correct?

Let $G=\mathbb Z / n \mathbb Z$ for $n > 2$ and let $G$ act on $\mathbb R^2$ linearly and effectively. Let $T_g (v)$ denote the element $gv$ where $v \in \mathbb R^2$. Assume that $\det T_g > 0$ ...
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1answer
36 views

My proof that $G(x)\to G / G_x$ is injective

Please could someone check my proof that $\varphi : G(x) \to G/G_x$ is injective? The notation is the following: $G$ is a group acting on a set, $G_x = \{g \in G\mid gx = x \}$ and $G(x) = \{gx ...
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1answer
117 views

Group Actions: Orbit Space

Given a group action $G\curvearrowright X$. Consider the orbit space: $\pi:X\to X/G$ Do continuous group actions correspond to open projections, i.e.: $$l_g\in\mathcal{C}(X)\quad(g\in ...
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1answer
31 views

A topological space with a transitive action.

Let $X$ be a topological space on which a topological group $G$ acts transitively. Given $x\in X$ let $$stab(x)=\{g\in G\;|\; gx=x\}.$$ I want to show that $X$ is homeomorphic to $G/stab(x)$ for any ...
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1answer
70 views

Group Actions: Discontinuity

Given a group action $G\curvearrowright X$. Then it need not be a continuous one: $l_g\notin\mathcal{C}(X)$ As an example I have in mind: $$k\in\mathbb{Z}:\quad l_k(x\in\mathbb{Z}):=x+k,\quad ...
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1answer
34 views

Symmetric Group acting on $X \times X$

The symmetric group $S_n$ acts on the set $X = \{1,\ldots,n\}$ and hence acts on $X \times X$ by $g(x,y) = (gx, gy)$. Determine the orbits of $S_n$ on $X \times X$. Not sure how do I actually ...
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0answers
33 views

Orbit space of action of a subgroup of a Lie group on a separable metric space

I am stuck on this question. Let $G$ be a Lie group acting freely on a separable metric space $X$. Assume that the orbit space $X/G$ is Hausdorff. Let $H$ be a normal Lie subgroup of $G$. Is the orbit ...
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3answers
171 views

Group acting on its subsets

Let $ G $ be a group with $ |G|=mp^\alpha $ where $ \alpha\geq1 $ and p is prime integer with $p \nmid m$. Then denote the set of subsets of G, having $p^\alpha$ size, with $X$. Then with the action ...
2
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1answer
82 views

Existence of a particular group action

Let $P$ be a group with normal subgroups $G$ and $H$, with $G \not \subset H$, $H \not \subset G$ and $G \cap H \neq 1$. Consider group actions $\theta : G \to Aut(H)$ and $\xi: H \to Aut(G)$ such ...
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28 views

Principal orbit type

I have trouble understanding the proof of Proposition 1.2.5 on p.17 in Audin's Torus Actions on Symplectic Manifolds: Let $G\curvearrowright M$ be a smooth action of a compact Lie group $G$ on a ...
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3answers
50 views

What is the relation between $\mathbb{C}[M]$ and $\mathbb{C}[M/U]$.

Let $M$ be a variety and let $U$ be a group. By definition, $M/U$ is the space of all $U$-orbits of $M$. Now we take coordinate rings $\mathbb{C}[M]$ and $\mathbb{C}[M/U]$. What is the relation ...
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1answer
50 views

Burnside's lemma simple use

Let's say that $D_3$ acts on a bracelet of 3 beads (Denote S), each bead can be Black or White. I want to count the number of different bracelets (4 - I believe) But using burnside's lemma I get ...
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2answers
43 views

Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
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1answer
42 views

Subgroups and an union of orbits

I have to prove or disprove the following statement: If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an ...
2
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1answer
48 views

Size of the orbits of a normal subgroup

So this is the question: Let $H$ be a finite subgroup of $G$, and let $(h,h')(x)=hxh^{-1}$ define an achtion of $H\times H$ on $G$, prove that $H$ is a normal subgroup of $G$ if and only if every ...
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1answer
23 views

$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Let $G$ be a p-group. $|G|=p^n$ for some n. Let X be a finite set so that $\,p\nmid |X|\,$, G acts upon X. Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$ I am trying to show ...
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1answer
29 views

$B$-action on $U$.

Let $G$ be an algebraic, $B$ Borel subgroup, and $U$ unipoent subgroup of $G$. For example, we take $G=GL_n$, $B$ the subgroup of lower triangular matrices, and $U$ unipoent upper triangular matrices. ...
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0answers
16 views

Group actions (congruence subgroups on integral binary quadratic forms)

I would really appreciate some help in computing the representatives for the space $Q_d/\Gamma_0(N)$ where $\Gamma_0(N) < \mathrm{SL}(2,Z)$ is the congruence subgroup at level $N$ and $Q_d$ is the ...
2
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1answer
43 views

Extending $*$-isomorphisms between $*$-algebras to cross products.

Let $G$ be a discrete countable group and suppose I have two $G$-$C^*$-algebras $A$ and $B$ such that there exists a $G$-equivariant isometric $*$-isomorphism $\varphi \colon A \to B$. One can extend ...
0
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2answers
61 views

S3 group action faithful?

I'm struggling with understanding the term "faithful". I read that a group action for example $S_3$ is faithful on {1,2,3}. Does that mean $S_3$ is not faithful on {1,2,3,4} because it never changes ...
3
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0answers
77 views

Motivation behind automorphism bases?

Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = ...
2
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1answer
43 views

What are all possible actions by automorphism of $H = \Bbb Z/3\Bbb Z$ on $N = \Bbb Z/6\Bbb Z$?

So the question is "What are all possible actions by automorphism of H on N?" with H = Z/3Z and N = Z/6Z. I completely guessed my way through how to go about solving this, but I started with finding ...