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Questions tagged [group-actions]

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 $2n$ act 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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$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H$. What kind of points have non-trivial stabilizer? And how many orbits are there?

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H= \{z \in \mathbb{C} | \Im(z) > 0 \}$ via Mobius transformation. $$ \text{ For } \gamma =\begin{bmatrix} a &b \\c&d \end{...
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1answer
22 views

Order of $C_{S_n}((12)(34)) \; \forall n\geq 4$ and its elements?

What is the order of $C_{S_n}((12)(34)) \; \forall n\geq 4$. Determine the elements of this centralizer explicitly Since $(12)(34)$ is a conjugacy class in $S_n \; \forall n\geq 4$, denote this ...
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0answers
11 views

Showing that left and right group actions of group $G $ on set $A$ induce the same equivalence relation on A?

Let $g.a$ denote the left action of group $G$ on a set $A \; \forall g\in G , a\in A$. Let $a.g$ denote the corresponding right action of $G$ on $A$. Then show that both induce the same equivalence ...
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1answer
27 views

Let $G=S_n$ and fix $i \in A=\{1,2,\ldots,n\}$ and let $G_{i}=\{\alpha \in G | \alpha(i)= i\}$. Prove $G_{i} \leq G$ and find $|G_i|$

I am not sure about $|G_i|$ but I was able to show $|G_i|$ as a subgroup as follow: Define $\; \;$ $\cdot$ $: G\times A \to A$ as $\alpha.i= \alpha(i) \; \; \forall \alpha \in G, \; i\in A$ Then $\...
2
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1answer
52 views

Shortest way to find the centralizer and normalizer of $A=\{1,s,r^2,sr^2\}\subseteq D_8$?

$A=\{1,s,r^2,sr^2\}\subseteq D_8$ We know $Z(D_8) = \{ 1,r^2\} \subseteq C_G(A) $ Hence $1,r^2 \in C_G(A) $ also $ sss^{-1} = s, \; s(sr^2)s^{-1} = sr^2 \; $ Hence $s \in C_G(A) $ Since $C_G(A)\leq ...
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1answer
25 views

Invariance under a normal subgroup

Let $G$ be a group acting on a set $A$. Let $N$ be a non-trivial normal subgroup of $G$. Suppose that $S$ is an $N$-invariant set, i.e. $n \cdot s \in S$ for all $s \in S$, $n \in N$. Must $S$ be $G$-...
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1answer
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“ Group action preserves the structure of vector space ” Can anyone prove or disprove it? [on hold]

" Group action preserves the structure of vector space " Can anyone prove or disprove it? I really can not see it. Can I take the vector space generated by {$1 , x, x^2$} over the field $F_3$? ...
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4answers
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Can I take the vector space generated by {$1 , x, x^2$} over the field $F_3$?

Let $V$ be the vector space over the field $F_3$ (field with $3$ elements) of dimension $3$. Define an action of the symmetric group $S_3$ on $V$ by $\sigma.e_i$= $e_{\sigma (i)}$ where {$e_1$, $e_2$,...
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1answer
29 views

Let $a \in G$. Show that for any $g \in G$, $gC(a)g^{-1} = C(gag^{-1})$.

Let $a \in G$. Show that for any $g \in G$, $gC(a)g^{-1} = C(gag^{-1})$. Note that $C(a) = \left \{g \in G : ag = ga \; or\; gag^{-1} = a\right \} $. How do I begin such a problem? I thought about ...
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1answer
32 views

Find the conjugacy class $O_{(234)}$ if $G=A_{4}$ is acting on itself by conjugation.

Find the conjugacy class $O_{(234)}$ if $G=A_{4}$ is acting on itself by conjugation. I calculated $(x)(234)$ for all $x\in A_{4}$ and got the set $O_{(234)}=\left \{(234), (143), (142), (123), (132)\...
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1answer
18 views

Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.

Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$. Can someone give me a head start to this problem? $X_{g}=\...
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votes
1answer
46 views

Why does the covering transformation group act properly discontinuously on the fiber?

In Step 2, I don't understand the part "for otherwise, two points would belong to the same orbit and the restriction of $\pi$ to $U_\alpha$ would not be injective". What two points? If $g\ne e$ and $...
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1answer
25 views

Application of Burnside's lemma to necklaces of three colors

I am trying to figure out the following problem using Burnside's lemma/formula: How many different necklaces can we make using 12 equally spaced stones if we have 4 red, 5 green and 3 blue beads? I ...
3
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1answer
35 views

Proving the number of commuting pairs of elements in $G$ equals number of conjugacy classes in $G$ times $|G|$

The first part of this problem asks to describe $\operatorname{Hom}(\mathbb{Z}^2,G)$ as a subset of $G \times G$ which turned out that $\operatorname{Hom}(\mathbb{Z}^2,G)$ is the set of pairs of ...
2
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1answer
29 views

$G_{x_1x_2+x_3x_4}\cong D_8$

Let $R=F[x_1,x_2,x_3,x_4]$ be the set of polynoms in 4 variables over a field $F$. Let a map $\varphi:S_4\to \operatorname{Sym}(R)$ by $$ \\ (f(x_1,x_2,x_3,x_4))\varphi(\sigma)=f(x_{1\sigma},x_{2\...
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0answers
31 views

Permutations on vertices of cubes and hence finding volume enclosed by the vertices

Denote $C$ to be the cube $C=\{(x_1,x_2,x_3)|0 \leq x_1,x_2,x_3 \leq 1\}$ and let $V=\{ (x_1,x_2,x_3)|x_1,x_2,x_3 \in \{0,1 \} \}$ be the set of vertices of the cube. Let $A=$convex$((0,0,0) , (1,0,...
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0answers
34 views

Equivalence of definitions of ergodic action

Let $G$ be a group acting on a probability measure space $(X, \mu)$ by measure-preserving transformations. I have read the two following definitions of ergodicity of such an action: For every ...
5
votes
1answer
32 views

Ergodic action of dense subgroup

Let $G$ be a group acting ercodically on a probability measure space $(X, \mu)$. Let $\Gamma$ be a countable dense subgroup of $G$. Is the action of $\Gamma$ also ergodic? The case I am interested in ...
3
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3answers
52 views

Group actions and orbits

Let $X=\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\}$ and $G=S_4$ (a) Show that $\phi \{x_1,x_2\}=\{\phi (x_1),\phi (x_2)\}$ determines an action of $S_4$ on $X$, where $\phi \in G$ (b) ...
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1answer
28 views

Group of covering Transformation corresponding to proper discontinuous action of a group on a connected space is the group itself.

Suppose $G$ acts properly discontinuously on a connected space $X$. Show that the group $G(X,p,X/G)$ of the covering transformation of $p:X \rightarrow X/G$ is $G$. I have tried in this manner- ...
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2answers
44 views

Determine number of conjugacy class in $D_8$

I’m using the formula that the number of conjugacy class is given to be $\frac{1}{|G|}\sum|C_{G}(g)|$, where $C_{G}(g)=\{h \in G ; gh=hg\}$, which is a special result by Burnside’s theorem. I found ...
1
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1answer
27 views

Set of Double Cosets

Consider a group $G$ with subgroups $H$ and $K$. The double cosets $\{ HgK : g \in G \}$ partition $G$. It seems this is just a set, not a group action. But I thought I might be missing something- can ...
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0answers
22 views

Continuous map from function space to quotient space maps through projection?

Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an ...
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1answer
30 views

Let H be a subgroup of finite group G. G acts on G/H by left multiplication. This induces a homomorphism. Show that its kernel is in H

Let G be a finite group and H is a subgroup of G. We have G acts on the set of left co-sets of H (G/H) by left multiplication x(gH)=xgH. This action induces a homomorphism from G to perm(G/H). Show ...
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0answers
25 views

Is there an algorithm to find relations in polynomial algebra?

In the context of equivariant cohomology for a GKM-manifold, I'm trying to compute the cohomology of the variety given by $x_1y_1+x_2y_2+x_3y_3=0$ with the action of $(\mathbb{C}^*)^3$: $$ ((t_1,t_2,...
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1answer
19 views

G-congruence equivalence relation

Let $G$ act on $X$. An equivalence relation ~ on $X$ is called a $G$-congruence if whenever $x,y \in X$ then $x$~$y$ implies $xg$~$yg$ for all $g \in G$. Suppose ~ is a $G$-congruence on $X$. Show ...
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0answers
21 views

invariant neighbourhood under a continuous group action

I found this statement and I am really struggling trying to come up with a proof of it. The situation is the following: Let $G$ be a compact topological group acting continuously on a compact ...
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1answer
71 views

Why is this sequence of isometries contained in a compact set?

I'm reading through a proof of the Mostow rigidity theorem (pages 738-740) in https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf but I'm stuck at a certain part. In Step 2, we consider the vertical ...
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2answers
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Applying Burnside's lemma to show there are $k+n-1 \choose k-1$ ways to store n stars in k bins?

Usually it is proven using multisets I think, but I wondered how Burnside's lemma could be applied. Everytime I tried to wrap it around my head the indices didn't seem to fit. So I TeXed it and ...
4
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1answer
89 views

Fundamental group of $T/\mathbb Z_2\setminus\{\text{singular points}\}$

Let $T=S^1\times S^1$. There is a $\mathbb Z_2$-action on $T$ defined by $x\sim -x$ (considering $T$ as a quotient of $\mathbb R^2$). The quotient $X:=T/\mathbb Z_2$this has four singular points, say ...
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1answer
40 views

number of orbits of $A_5$ acting by left multiplication in $S_5$

Looking for a very fast/"smart" way to compute this number (it was a question asked on an hour-long exam I recently took, so listing everything out for each element in $S_5$ was not an option since I ...
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1answer
18 views

Setwise stabilizer and relationship with pointwise stabilizer

Let $ T\subseteq S$ denote a subset of $S$ and $G$ denote the group acting on $S$. Let $$G_T=\bigcap_{s\in T} G_s \quad\text{and}\quad G_{\{T\}}=\{ g\in G: T = T^g \} $$ represent the pointwise ...
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0answers
29 views

Effective epimorphisms in $G$-set

$\newcommand{\gset}{G\text{-}\mathsf{set}}$ Let $G$ be a group and $\gset$ the category of $G$-sets, whose morphisms are $G$-maps (i.e. set maps where the map commutes with $G$-action). What are the ...
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1answer
48 views

Fibred products and epimorphisms in $G$-set

$\newcommand{\gset}{G\text{-}\mathsf{set}}$ Let $G$ be any group and $\gset$ be the category of $G$-sets, with morphisms being $G$-maps. That is an object of $\gset$ is a pair $(X,\rho)$ where $\rho:G\...
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0answers
15 views

Lifting of operator under free action

Suppose a Lie group $K$ acts freely, properly and isometrically on a Riemannian manifold $(M,g)$, which is equipped with a $K$-invariant measure. Then $(M/K,\check{g})$ is a Riemannian manifold, where ...
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1answer
45 views

How to show the following about the infinite group?

Prove, or disprove the statement: An infinite group $G$ can never act transitively on a finite set $X$. What i know is that a group action $G \times X \to X$ is transitive if it possesses only a ...
2
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1answer
29 views

Minimum size of index of a proper subgroup of a finite, non-abelian simple group $G$

Let $G$ be a finite non-abelian simple group and $p$ the largest prime divisor of $|G|$. Show that if $H < G $ then $|G : H | \geq p $. This is from a chapter of a book about group actions, ...
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0answers
19 views

Showing a group generated by reflections of the side of a hexagon is a discrete action on the Euclidean plane.

How would one go about showing that the action of the group generated by reflections with respect to the sides of a regular hexagon in the euclidean place, acts discretely on the euclidean plane.
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0answers
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Number of k-combinations of an n-set using orbit-stabilizer theorem

I have two similar questions: Justify that a set of n elements has $\binom nk$ subsets of k elements by using the orbit-stabilizer theorem. The orbit-stabilizer theorem says that $|Orb(x)|=[G:Stab(x)...
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0answers
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No. of representatives of the elements of $Bw_{\sigma} B$ in $GL_3(\mathbb F_q)\setminus B$ (the set of all right cosets of $B$)

Let $K=\mathbb F_q$. Let $B$ be the set of all upper triangular matrices in $GL_3(K)$. For $\sigma \in S_3$, let $w_\sigma := \begin{pmatrix} e_{\sigma(1)} & e_{\sigma(2)} & e_{\sigma(3)} \...
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0answers
28 views

Confirmation on $2\times 2\times n$ Rubik's cube group

I believe that I have come up with formulae for the groups representing the $2\times 2\times n$ Rubik's "cubes," but I need someone to confirm that they are correct. Here are the groups that I came up ...
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1answer
20 views

The induced map $f' : S^3\times S^3\rightarrow SO(4)$ of $f : S^3\times S^3 \rightarrow GL_4(\mathbb{R}) $ is a homomorphism.

I am wanting to Show that a map $f' : S^3\times S^3\rightarrow SO(4)$ induced from a map $f : S^3\times S^3 \rightarrow GL_4(\mathbb{R}) $ is a homomorphism where $S^3$ is a Lie group of unit ...
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1answer
37 views

The space of oriented 2-dimensional subspaces of $\mathbb{R}^4$ is compact and connected.

I am trying to Show that the set of oriented 2-dimensional subspaces of $\mathbb{R}^4$ is 4 dimensional compact and connected homogeneous space. The given hint is that (1) $SO(4)$ is smooth ...
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1answer
18 views

Action of $SO(4)$ on $M$ that is transitive

I am trying to find a transitive left action of $SO(4)$ on the set $\mathcal{P}$ of oriented 2-planes in $\mathbb{R}^4$. Each element in $P$ in $\mathcal{P}$ is a 2 dimensional subspace of $\mathbb{...
2
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0answers
27 views

Help understanding the permutation group action .

I'm trying to understand actions in regards to group theory . specifically in my notes I found the following example : Say G=$A_4$, for $x \in \{1,2,3,4\}$, and $\tau \in A_4$ We let $x^{\tau}$ be ...
2
votes
1answer
74 views

On the set of cusps $\mathbb{P^1(Q)}$

The following is from the book Modular Forms by W Stein: My questions: $1-$ The very same book defines a cusp form as a modular form when $f(\infty)=a_0=0$. Is the set of cusps a different ...
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0answers
32 views

Action of SL2 on subspaces over a finite field

Let V a vector space of dimension 2 over $\Bbb F^3$, i.e $V = \Bbb F^2_3$. We know that $SL_2(\Bbb F^3)$ acts on $\Omega = \{W \le V : \dim(W) = 1\}$. Note that $card(\Omega)= 4$. It implies that ...
1
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1answer
11 views

Compute the number of distinguishable coloured wheels having 12 compartments that can be formed with $q$ colours

Compute the number of distinguishable coloured wheels having 12 compartments that can be formed with $q$ colours.(Assume that only one colour can be used on a single compartment and that the same ...
1
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1answer
34 views

Fixed set is a block

Let $G$ act transitively on a set X and let $x\in X$. Prove that the fixed set $X^{G_x}$ is a block of $X$. Deduce that if G acts primitively, then $X^{G_x}={x}$ or else $G_x=\{1\}$ and $X$ is a ...
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0answers
52 views

Transitive action and elementary abelian 2-group of stabilizer

Show that no subgroup $G$ of $S_5$ acts transitively on $X=\{1,2,3,4,5\}$ in such a way that $G_x$ is an elementary abelian 2-group for any $x\in X$ I want to solve this problem. I decided to use ...