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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Existence of dense subsets $G$-invariant.

Let $G$ be a group that acts on a manifold $X$. It is well know that the orbir space $X/G$ isn't in general a manifold. But how can I prove that there is always a dense open $U \subset X$ that is ...
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2answers
119 views

Using counting formula to get |G| = |kernel φ||image φ|

The counting formula I am saying : Let S be a finite set on which a group G operates, and let Gs and Os be the stabilizer and orbit of an element s of S. Then |G|=|Gs||Os| or ...
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1answer
144 views

Classification of transitive G-sets for a given group of small order

Given a group of small order (<30), how does one go about systematically finding all the transitive G-sets up to isomorphism? By X and Y being isomorphic we mean there are maps $f:X \rightarrow Y$ ...
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0answers
94 views

Stabilizer map on transitive G-set defines a morphism with G acting on subgroups by conjugation

This is part of a homework problem for a graduate course on abstract algebra. Given a transitive G-set $X$, show that the map that assigns to $x \in X$ its stabilizer defines a morphism of G-sets ...
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1answer
99 views

Is the kernel of this group action the centralizer?

In Dummit and Foote, they state "... let the group $N_G(A)$ (normalizer) act on the set $A$ by conjugation. It is easy to check that the kernel of this action is the centralizer $C_G(A)$." From ...
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132 views

Why are they called orbits?

When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\} $$ that are called orbits. Although, the only reason I find convincing for that name is that in ...
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1answer
24 views

Modules over a group presented via a free group.

Say $G$ is presented via a free group $F$ freely generated by $S=\{s_i, 1=1,2,\dots\}$. Then $\pi:F \rightarrow G$ the canonical projection. Let $R$ be any commutative ring. Can we follow that any ...
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1answer
133 views

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$.

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$. The element $m$ is $ \left( \begin{matrix} 2 & 1 \\ 0 ...
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1answer
129 views

How to define own group action in GAP?

I am beginner in GAP. I have a group and a set. I wish to define an action the group on the set in my own way and wish to calculate its orbits and stabilizers. Is it possible? What is process?
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3answers
166 views

Action in Groups (transitively)

Let X be a set of order n. a) If G acts transitively on X then n divides $| G |$. b) If G acts 2-transitively on X then n(n-1) divides $| G |$ For a) i first prove that if G acts transitively on ...
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49 views

Smooth group actions ----> “Action Lie groupoids”?

It is well known that any group action of a group G on a set X gives rise to the corresponding action groupoids, see http://math.ucr.edu/home/baez/week249.html , for instance. Now in a perfect ...
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2answers
121 views

Finding the kernel of an action on conjugate subgroups

I'm trying to solve the following problem: Let $G$ be a group of order 12. Assume the 3-Sylow subgroups of $G$ are not normal. Prove that $G\cong A_4$. Here's my attempt: let $\mathscr S$ be ...
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1answer
117 views

The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
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0answers
195 views

Prove that a doubly transitive group is primitive.

My Attempt: (a) $S_n$ is transitive on $\{1,2,\dots,n\}$ and for any $(i,j)\in G_a,~(i,j)i=j$ whence $G_a$ is transitive on $\{1,2,\dots,n\}-\{a\}.$ (b) Without loss of generality let $|A|\ge2.$ ...
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5answers
218 views

What does it mean for a group to “act algebraically”?

I'm reading the paper How to use finite fields for problems concerning infinite fields, by Serre. In Theorem 1.2 on page 1, he says Let $G$ be a finite $p$-group acting algebraically... In the ...
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1answer
387 views

left regular representation(group action) of a finite group $G$ Dummit Foote 4.2.11

Let $G$ be a finite group and let $\pi : G\rightarrow S_G$ be the left regular representation. Question is to : Prove that if $x$ is an element of order $n$ and $|G|=mn$ then $\pi(x)$ is a product ...
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1answer
134 views

$Q_8$ is isomorphic to a subgroup of $S_8$ but not to asubgroup of $S_n$ for $n\leq 7$.

Question is to Prove that : $Q_8$ is isomorphic to a subgroup of $S_8$ but not isomorphic to a subgroup of $S_n$ for $n\leq 7$. I see that $Q_8$ is isomorphic to subgroup of $S_8$ by left ...
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0answers
108 views

Fundamental group of a space under a group action

The short version: Why does a Borcea-Voisin threefold has trivial fundamental group? Well, what is a Borcea-Voisin threefold? These are named after Borcea and Voisin, who introduced a method of ...
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1answer
70 views

Triviality of the principal fiber bundle obtained from quotienting a manifold by a free and proper action

Let $M$ be a smooth manifold on which acts the $G$-action $\Phi$. According to the quotient manifold theorem, if $\Phi$ is free and proper, then the orbit space $M/G$ has a unique smooth manifold ...
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2answers
119 views

Diagonal groups and semidirect products

I am studying a text on permutation groups, which has the following example in a section on regular normal subgroups: If $Z(N)=1$, then $N \cong \mathrm{Inn}(N)$, the group of inner ...
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3answers
46 views

Prove that $ \pi_g $ is a permutation

Suppose that $ G $ is group and $ g $ is any element from $ G $ ($g \in G $). We define a function $ \pi_g : G \rightarrow G,\ \ \pi_g(x) = g \cdot x$. Prove that $ \pi_g $ is a permutation of the set ...
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1answer
121 views

A group action contains another action as a normal subgroup

I have a question on what it means for one group action to contain another group action as a normal subgroup. I am guessing this means that the image of the first group action contains the image of ...
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1answer
175 views

Left regular action isomorphic to the right regular action

I am reading something on regular groups and I have a question on why the left and right regular actions are isomorphic. Let $G$ be a group. Consider the homomorphism $\rho: G \rightarrow Sym(G), g ...
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1answer
333 views

Number of Orbits in Group Action

Let G be a group of order 15 acting on a set of order 22. Assume there are no fixed points. Determine how many orbits there are.
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2answers
106 views

$G$-set terminology

When a group action $G \times X \rightarrow X$ is defined with a group $G$ and a set $X$, why is there not a special name for the set $X$? I know that this is referred to as a $G$-set, but the set $X$ ...
3
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2answers
66 views

action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
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0answers
45 views

boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on ...
11
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1answer
281 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
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67 views

Tangent bundle of a quotient manifold

I am interested in the tangent bundle of the quotient of a manifold by a proper and free action but I can't find any reference on the net. Does anyone know a book or article where it is described ?
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43 views

When the $(\mathbb R,+)$-action defined by the flow of a vector field on a manifold is proper ?

I'm interested in the conditions that a vector field has to satisfy in order for its flow to define a proper action. Technically, let $\xi$ be a smooth vector field. For each $m\in M$, there is a ...
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1answer
111 views

Kernel of this action is trivial?

Let $S_n$ act on $S_n/H=\{ H,t_2 H,...,t_e H\}$ from the left and let $n\geq 5$, $e \geq 3$. Then the kernel of this action is trivial. This supposedly follows because $e \geq 3$, but I haven't the ...
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0answers
45 views

Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
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1answer
116 views

Help with Conrad's “Recognizing Galois Groups”

I'm trying to do the proofs of theorems 2.1 and 2.2 in this article by Keith Conrad. I'll just quote the parts of theorem 2.1 I have problems with: if $a \sim b$ then $ga \sim gb$... if $a \neq b$ ...
2
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1answer
55 views

fundamental group of a graph is free

Let $X$ be a connected graph, and $T$ its maximal tree. Via covering spaces and deck-transformations, how one can prove that $\pi_1(X)= \pi_1(X/T)$?
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71 views

Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
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1answer
87 views

Splitting field of resolvent equals that of $f$

Lemma: Let $\Psi \in k[X_1,...,X_n]=:B$ be s.t. $stab_{S_n}(\Psi)=H \subset S_n$, $S_n/H=\{ \Psi, t_2 \Psi,..., t_e\Psi \}$, $\Delta_\Psi$ the discriminant of $L_\Psi:=\prod_{i=1}^e (X-t_i \Psi)$, and ...
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1answer
52 views

Partitions and Orbit Sizes

If $U,V \subset S_n$ are subgroups with $S_n//U = \{id,g_2,...,g_e\}$ and $\alpha_j$ is $\frac{1}{j}$ times the number of $i\in [e]$ s.t $[V:V \cap g_i U g_i^{-1}]=j$ then $(\alpha_1,...,\alpha_e)$ is ...
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1answer
75 views

Show that the stabilizer is a prime subgroup

We define a subgroup $H$ as being convex if $g\in H\implies h\in H$ for all $1\leq h\leq g$. A convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that ...
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0answers
67 views

Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
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1answer
34 views

Quotient by the action of positive reals

Today in my complex analysis class Riemann sphere was defined, and of course the construction caused questions, such as "why don't we distinguish between all the various infinities?" and "Would it ...
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0answers
103 views

Degree of factor in a resolvent

Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma ...
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2answers
90 views

Faithful group actions and dimensions

Just a quick question. I'm trying to understand the answer to one of my previous questions. The precise problem I want to show is as follows. Let $G$ be a group acting faithfully on a manifold ...
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0answers
95 views

trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...
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0answers
112 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
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0answers
106 views

Group action on a manifold with finitely many orbits

I'm looking for a result along the lines of the following: Let $G$ be a group acting on a set $X$. If the action partitions $X$ into finitely many $G$-orbits, then $\dim G \geq \dim X$. For ...
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1answer
429 views

Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
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1answer
69 views

Are subgroups kernels of some $1$-cocycles?

If an $1$-cocycle $\sigma :G\to K$ has image in a group on which $G$ acts, then the set of element of $G$ that is mapped to $\text{$e_K$}$ by $\sigma$ forms a subgroup of $G$. This could be proved by ...
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1answer
67 views

Group acting by isometries on a length space

I am reading the book A course in metric geometry by Burago, Burago and Ivanov. I have some difficulties with an exercise 3.4.6 on page 78. The exercise is the following: Let a group $G$ act by ...
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2answers
103 views

Proving those elliptic matrices in $\operatorname{SL}_2(ℤ)$ are not conjugate

Set $\mathbf{\Gamma} = \operatorname{SL}_2(ℤ)$, let $\mathbf{H}$ denote the upper half plane. and let $$\Gamma_0 (N) = \left\{ \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] ...
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1answer
38 views

What subsets of a covering space cover their image?

Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map? If it makes things easier, assume $E$ is simply connected, that is, the ...