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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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Differential Action of Möbius Transformations

The group $\mathrm{PSL}_2(\mathbb{R})$ acts on $\mathbb{H}$ via Möbius transformations, that is \begin{align*} g=\begin{pmatrix} a & b \\ c & d\end{pmatrix}:z\mapsto \frac{az+b}{cz+d}. \end{...
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Consider G = S4 with the action of conjugation. [on hold]

Find the orbits of $S_4$ under this action. For each orbit choose a representative element and find its stabilizer subgroup. Write down the class equation of this action in its set theoretic and ...
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Group action by conjugacy and “curves”

Let $G$ be a group, $H\unlhd G$ and "$\cdot$" the $G$-action by conjugacy on $H$: \begin{alignat*}{1} \cdot:G \times H &\longrightarrow H \\ (g,h) &\longmapsto g \cdot h:= g^{-1}hg\\ \end{...
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Does a finitely generated group $G$, ever act on $G/N$ freely?

Say that $G$ is a finitely generated group on $k \geq 2$ generators and $N$ is a normal subgroup of $G$. I want to know if I can construct a $G$-action on $G/N$ such that the action is free. None of ...
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Group action $\phi: G \times X \rightarrow X$ of group $G$ generated by elements of order $7$ and $11$ on $X$ with $|X|=8$ is not transitive

Let $G=\langle a,b \rangle$ with $ord(a)=7, ord(b)=11$ $|X|=8$ Show that the group action $$\phi: G \times X \rightarrow X$$ $$(g,x) \mapsto g.x$$ is not transitive I know that $\phi$ is ...
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Manifold acted on by every finite group

Is there a (second-countable, connected) manifold that admits a faithful continuous group action by every finite group?
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Number of $S_n$-orbits in $P^k(\{1,\dots,n\})$

Let $n$ and $k$ be integers with $n\ge1$, $k\ge0$, and let $a(n,k)$ be the number of orbits of the symmetric group $S_n$ on the $k$-th iterated power set $$ P^k(\{1,\dots,n\}) $$ of the set $\{1,\...
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Linear action of group $G$ on vector space $V$

Let $p$ be a prime number, and let $G$ be a finite group whose order is a power of $p$. Let $F$ be a field with characteristic $p$, and $V$ a nonzero vector space over $F$ equipped with linear action ...
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For a finite group $G$ consider the left action of conjugation of $G$ on itself $·:G×G→G$ given by $g·x=gxg^{-1}$.

For a finite group $G$ consider the left action of conjugation of $G$ on itself $·:G×G→G$ given by $g·x=gxg^{-1}$. Prove that if $H$ is normal in $G$ then $H$ is the disjoint union of orbits of this ...
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Prove that this group action is continuous from $S\times\Bbb{R}^2\to\Bbb{R}^2$

Let me give some description about the notations used- S denotes the collection of all transformations on $\Bbb{R}^2$ by integer coordinates. i.e. $S=\{t_v|t_v(x)=x+v,\forall x\in\Bbb{R}^2; \forall v\...
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Cotangent lift of left translation SE(3)

I am wondering how I can verify the cotangent lift of left translation $ T^*L_{(\Lambda,\phi)}:T^*_{(\Lambda,\phi)}G\to T^*_eG$ which reads $$ (\Lambda,\phi)^{-1}(\alpha_\Lambda,(\phi,v))= T^*L_{(\...
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Compact group acting on regular space

Let $G$ be a compact topological group, $X$ be a regular topological space. Then the quotient space given by the continuos action of $G$ on $X$, $X/G$ is also regular. Here's my attempt, though I feel ...
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Number of conjugates of $(12)(34)(56)(789) \in S_{10}$

Number of conjugates of $(12)(34)(56)(789) \in S_{10}$ This is how I calculated it and got $840$ as a result: $${10\choose 2}\cdot{\frac{2!}{2\cdot3!}}\cdot {8\choose 3}\cdot\frac{3!}{3}$$ What I ...
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How $G = D_n$ act on $\mathbb{C}^3$?

In particular, given $G = D_n = \langle \kappa, \rho \rangle$, I was wondering what the $3\times 3$ matrix representing $\kappa$ would look like for the action of $G$ on $\mathbb{C}^3$. For this ...
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1answer
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Euler characteristic expression in terms the number of fixed points of an $\mathbb{S}^1$ action

I have found in a paper* that I am reading that Given $(M,J)$ compact (smooth) manifold with an almost complex structure $J$, if we have an $\mathbb{S}^1$ action with isolated fixed points then $ \...
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Intersection number via tangent spaces

Assume that finite groups $G_1$ and $G_2$ act smoothly on a manifold $M$ in such a way that the fixed point set, $M^{G_1\cap G_2}$, is an oriented closed manifold, $M^{G_1}$ and $M^{G_2}$ are its ...
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1answer
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Right action induced by free group homomorphism

I was reading this proof by Steinberg of Nielsen-Schreier theorem and i had a doubt about proposition 1, that basically is an alternative universal property of free groups. It says: Let $X$ be a set ...
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Different ways we can paint the faces of a tetrahedron using $4$ colors

Suppose we want to paint the faces of a tetrahedron using $4$ different colors, assuming that we allow different faces to be painted with the same color. By not taking the symmetries of tetrahedron ...
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1answer
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Left coset as group action on a subgroup

Take a group $G$ and a subgroup $H$. Let's define the set of left cosets of H as: $$S=\{aH | a\in G\}$$ It's pretty clear to me we can define a left action from G to S like this:$$g(aH)=gaH$$ This ...
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Stabilizer in the definition of a Gelfand pair

I am following the textbook Representation Theory of the Symmetric Groups, by Tullio C.-S., Fabio S., and Filippo T., and am confused at the definition of a Gelfand pair. The definition is: Let $G$ ...
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A finite open cover for real semisimple Lie groups

Let $G$ be a real semisimple Lie group. Is it true that there always exists a finite open cover $\{\mathcal{O}_i\}_{1\leq i\leq n}$ such that $\mathcal{O}_i$ is invariant under the conjugation action ...
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Free action of a subgroup's normaliser on a set of orbits

I'm struggling with this question: Let $G$ be a group that acts freely on a set $X$, and $K < G$. Prove that the normaliser of $K$, $N(K)$, acts freely on $X/K$. The question was posed to me ...
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Number of orbits and representatives of an action of $Gl_2(\mathbb{Z}_2)$ in $M_2(\mathbb{Z}_2)$ by conjugation

So im asked to find the number of orbits and representatives of that action, my idea was to find all the possible rationals forms induced by a polinomyal of order 2 in $\mathbb{Z}_2[x]$, thinking of ...
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1answer
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The action of the group $\Gamma=\mathbb{Z}$ on the manifold $\mathbb{C}^n-\{0\}$

Let $\Gamma=\mathbb{Z}$ be the additive group of integers and give it the discrete topology. Suppose $\Gamma$ acts continuously on the topological n-manifold $\mathbb{C}^n-\{0\}$ by the map $x \mapsto ...
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Verifying the calculation of the number of conjugate elements

I would like to verify my understanding: Consider action of $S_7$ on itself by conjugation. I'm trying to compute: $Stab_{S_7}((1 2))$ $Stab_{S_7}((1 2 3 4 5 6 7))$ $Stab_{S_7}((1 2 3)(4 5 6))$ I'm ...
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39 views

Proving that $|S_8 \cdot a|=\binom{8}{4}\frac{4!}{4}$

So I came across with the following exercise: How many elements are conjugated with $a=(3\ 5\ 1\ 6)$ in $S_8$? The solution was: we are looking for $|Stab(a)|=|\{b\in S_8 : bab^{-1} = a\}|$ ...
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Computing $| \operatorname{Stab}_{S_n}(x)$| [duplicate]

Consider $s\in\{0,\dots,n\}$. Let $X$ be the set of all the $s$-subsets of $[n]$. Also consider the action of $S_n$ on $X$ by $\sigma\cdot \{x_1,...,x_s\}=\{\sigma(x_1),...,\sigma(x_s)\}$. Let $x\in X$...
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Calculating the orbit of a group

In the $S_7$ group, in action on itself by $xy=xyx^{-1}$, I would like to calculate the orbit of $(123)(456)$. I read the definition: Consider a group $G$ acting on a set $X$. The orbit of an ...
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Calculation $Stab_G$ of the group $G$

So, I was trying to understand the "Group action" theory. I read the definition of $Stab_G$ and I was trying to solve some basic questions. I came across with the following question: Let $S_7$ be ...
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Cellular action on CW complex

Let $G$ act cellularly on a CW complex $X$. For each $n\ge 0$, the action induces an action on the indexing set $I_n$ for the $n$-cells. Now look at the cellular chain complex $C_\bullet(X)$. Each ...
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1answer
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Prove that for $\pi_g(\alpha)=\alpha g$, $\pi$ is a group action

Let $R$ a principal ideal domain and let $G$ the group of all the invertible members of $M_2(R)$. Let $\Omega :=R^2$. For all $g\in G$ let $\pi_g:\Omega\to\Omega$ defined by $\alpha \pi_g=\alpha g$ ...
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Action of $GL(\Bbb F_2^3)$ on the sub-spaces of $\Bbb F_2^3$ of dimension $2$

Let $V=\Bbb F_2^3$ and let $G=GL(V)$ act naturally on the set $X=\{W\subset V:\text{sub-vector space,}\dim =2\}$ If $W\in X$ how do you determine the $Stab_G(W)$? and why shoud the cardinality of $...
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$G$-Action on a Ring extends to a Module

Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$. Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well ...
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How to show a certain group element must belong to the stabilizer of a set element

I'm studying for personal fun and culture group theory, specifically the orbit stabilizer theorem. I've then found this interesting problem: Let a group $G$ acting on $\Omega$ and take $\alpha,\...
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Group Acting on Variety

My question refers to some comments occured in following thread: Galois morphism - group acting on the variety The setting is that we have a finite Galois morphism $f: X \to S$, where $X$ and $S$ ...
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1answer
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Orbits of the conjugation action of $GL_3 (\mathbb{R})$ on the nonsingular symmetric $3\times3$-matrices

Let $S$ be the space of all symmetric $3 \times 3$ matrices of full rank and with real entries. $GL_3 (\mathbb{R})$ acts on this space by conjugation, \begin{align*} g.A = (g^{-1})^T A g^{-1}, \quad ...
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Orbifold Subchart Definition

I am currently reading Zvonkine's "An Introduction to Moduli Spaces of Curves and Their Intersection Theory" and I am hoping that someone here would be willing to clarify some aspects of his ...
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How orbits of $G$-sets and these characters are related

I've been learning about induced representations recently and I've come across something which I'm very confused about; For any $G$-set $X$, the number of orbits is equal to $(1_G, \chi_{\mathbb{C}[...
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If $|G|<\infty$ acts transitively on a set $X$ with $|X|=10$ then $G$ has an element of order $5$

Let $G$ be a finite group that acts transitively on a set $X$ with $|X|=10$. Show that $\exists g\in G$ of order $5$ There is a homomorphism $\phi:G\mapsto S_{10}$ that sends $g\in G$ to its ...
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Centralizer and Normalizer as Group Action

Could someone explain the following to me? I understand all the terms used in the text but have no intuition of what it's saying. Dummit and Foote Pg. 52
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A certain colimit of representables sheaves, namely group actions, is a sheaf: why?

In the first answer to this post on MO, one finds that When you look at the category of sheaves on the category of finite action with the natural topology (covering are surjection of finite ...
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How to prove $\text{Sh}_G(X)\simeq \text{Sh}(G\backslash X)$ when $X$ is a free $G$-space?

Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which act on $X$ continuously from the left....
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2answers
398 views

What is the relationship between the orbit-stabilizer theorem and Lagrange's theorem?

Is Lagrange's theorem used to prove that the length of the orbit times the order of the stabilizer is the order of the group, or is Lagrange's theorem a corollary of the orbit-stabilizer theorem?
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Let $G$ be transitive on $S$. Show that the action is primitive if and only if every $\operatorname{Stab}_G(a), a\in S$, is a maximal subgroup of $G$.

I am self-studying "Classical Groups and Geometric Algebra" by Larry C. Grove. This is the 2nd question of the exercises of the 0th Chapter. Let $G$ be transitive on $S$. Show that the action is ...
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A question about “$\times$” notation with group actions

Per Wikipedia, a (left) group action is defined as follows: If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$ is a function $$\varphi : G \times X \to X : (g,x)\...
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1answer
56 views

Group Action on a Scheme

Let $X$ be scheme and $G \subset Aut(X)$ be a subgroup of automorphism group of $X$. By definition $G$ acts espectially on local sections $\mathcal{O}_X(U)$ for open $U$ and one can therefore define ...
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Galois Action on Scheme

Let $X$ be a $K$-scheme and $L \vert K$ be a Galois extension with Galois group $G= Gal(L,K)$. Let consider the base change $X_L := X \otimes_K L:= X \times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-...
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1answer
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Find a set of representatives of cosets in $S_4$

Problem: Find a set of representative of the left cosets of $H$ in $S_4$, where $H = \left\{ \sigma \in S _ { 4 } \mid \sigma ( 4 ) = 4 \right\}$. Solution: In the case of $S _ { 4 }$ we have $\{ \...
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1answer
51 views

Involution action on $H^3(S^1\times S^2)$

I am studying about involution action $I^*$ on de Rham cohomology group $H^3(S^1\times S^2)$ induced from an action $I\cdot (z,x)=(\overline{z},-x) $ where $S^1\times S^2\subset \mathbb{C}\times \...
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1answer
27 views

Representatives for all conjugacy classes of elements of order 15 in A11

I am trying to find representatives for all conjugacy classes of elements of order 15 in $A_{11}$. It is not hard to see that $(12345)(678)$ and $(12345)(678)(9 10 11)$ are the representatives for ...