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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Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?

Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...
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2answers
45 views

For a given group $G$ , what are the sets on which a non-trivial group action of $G$ can be defined ?

Say we are given a group $G$ , we want to find those sets on which we can define an action of $G$ ; now in this sense any set $X$ works as we can always define the trivial action $o:G \times X \to X$ ...
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1answer
13 views

Show that the following group action has a non-zero singleton orbit.

Let F be a finite field of characteristics prime p.Let G be a group of order $p^r$ for some r.Let G acting on $F^n$ for n>1.Then show that there exist a non-zero vector in $F^n$ whose orbit will be ...
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21 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
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32 views

The action of a Lie algebra on a manifold is a Lie algebra homomorphism. How to show it?

By definition, the action of a Lie algebra $\mathfrak g$ on a manifold $M$ is a Lie algebra homomorphism, $\mathcal A: \mathfrak g\rightarrow\mathfrak X(M), \xi\mapsto\xi_M$ such that the action map ...
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1answer
22 views

Why the homomorphism from g acting on a to left coset of stabilizer of a is surjective?

Suppose $b = g \cdot a$. Then $gG_a$ is the left coset of $G_a$. The map $b = g \cdot a \rightarrow gG_a$ is a map from $C_a$ to the set of left cosets of $G_a$ in $G$. Dummit says this map is ...
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21 views

Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
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55 views

Slice at a point of a topological space

The definition is from the following link -Slice at a point of a topological space Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of is called a slice at a ...
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1answer
32 views

How to write isomorphism classes of group actions.

I need to find all of the isomorphism classes of transitive actions for $\mathbb{Z}_{4}$. I know that they are in a bijection with the conjugacy classes, and that since the group is abelian then ...
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15 views

When does a homogeneous space define a fibration?

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we ...
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1answer
22 views

Character of the algebra $\mathbb{C}[G]$ as $G \times G $-module

Let $G$ be a finite group. We can define an action of $G\times G$ on the group algebra $\mathbb{C}[G]$ in the following way: If $x \in \mathbb{C}[G]$ then $(g,h)\cdot x=gxh^{-1}$. Now, what about ...
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26 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
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17 views

About Galois Covering Theory

so I am studying somethings about Galois Covering and I am writing a beamer to present for my friends of the university. But I would like of somethings about the author of Covering Galois Theory to ...
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2answers
59 views

Isomorphisms of two subgroups in $S_6$

Is there any nice group action to see why groups $S_2 \times S_4$ and $S_2 \wr S_3$ of order 48 are isomorphic? Or is this "just" an abstract property which becomes invisible when we switch to ...
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1answer
31 views

Can kernel of homomorphism tell you when a group action cannot be constructed?

I understand that for every action of group $G$ on a set $X$, there is a homomorphism: $$G\rightarrow S_X$$ It seems to me that this can be used to rule out many possible actions. For example, a group ...
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1answer
38 views

Action on $G$ by inner automorphism

I wonder something about an action of a group $A$ on a group $G$ by a automorphism; There are many nice result related with some restrictions such as when $(|A|,|G|)=1$ , $G$ is abelian or ...
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1answer
44 views

Understanding Group Action

In general are we just supposed to make an educated guess about what the $G$-set is for a group action is if it's not specified? Here are two examples of what I mean. I am asked to find a fixed ...
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2answers
58 views

Group Action problem - Proving an action and finding number of orbits

The following question is from Serge Lang's Undergraduate Algebra. I have trouble in finishing part a and understanding part b. Let $S, T$ be sets and let $M(S,T)$ denote the set of all mappings ...
2
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1answer
43 views

How general is the class equation?

I generally see the class equation stated for the action of a group on itself by conjugation as follows where $C(G)$ is the centralizer of $G$: $$|G|=|C(G)|+\sum{sizes\ of\ nontrivial\ conjugacy\ ...
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1answer
60 views

Functor between two categories

Let $G$ be a finite group. Let $C$ be the category with objects subgroups of $G$ and morphisms between two subgroups $H,H'$ be $ \{ g \in G \mid g H g^{-1} \subset H ' \}$. Let $D$ be the category ...
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1answer
37 views

Group Action and Orbits

I am looking at the following example which says find the orbit of $0$ under addition by $2$ and $3$ if $\mathbb{Z}_4$ acts on itself by addition. So to find the orbit of $0$ we are looking at the set ...
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1answer
18 views

Free and proper action

I don't know how to solve this problem. Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by ...
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2answers
78 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
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1answer
28 views

$AGL(V) = V \rtimes GL(V)$ with $GL(V)$ acting from the right

For a vector space $V$, I have constructed $AGL(V) = V \rtimes GL(V)$ as the elements $(v, A) \in V \times GL(V)$ (Cartesian product of sets, not a direct product) with multiplication $(v, A) (w, B) = ...
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20 views

Symmetric groups and transitive action

I am trying to show that for $(x,x_1),(y,y_1)$ there exists $g\in S_n$ such that $gx=y$ and $gx_1=y_1$ where $x$, $x_1$, $y$, $y_1\in \{1,2,3,\dots ,n\}$ and $x\neq x_1$, $y\neq y_1$. Is this claim ...
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1answer
51 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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1answer
24 views

Transitive action on Poincare upper half plane

I am trying to prove that the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ via $$ \left( \begin{array}{ c c } a & b \\ c & d \end{array} \right)z\rightarrow \frac{az+b}{cz+d} $$ ...
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0answers
30 views

Homomorphism from a subgroup to a group is injective.

I'm reading a proof and I don't quite understand one step of the proof. We want to deduce that if G acts transitively on A then $ \bigcap_{\sigma \in G} \sigma G_{a} \sigma^{-1} = 1$. (Where $G_{a}$ ...
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29 views

Free action of symmetric group

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$? Is there a compact manifold which can be act freely by all symmetric ...
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12 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
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3answers
79 views

Involution on Cantor space with exactly one fixed point

Let $X=\{0,1\}^{\mathbb{N}}$ be the Cantor space. What is an example of a continuous map $\sigma : X \to X$ with $\sigma^2=\mathrm{id}$ and $\# \{x \in X : \sigma(x)=x\} = 1$? This has to exist, ...
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28 views

Help with terminology

I need some help unraveling the terms that appear in the following passage. I found it in a book on some conference proceedings related to Differential Geometry. Let $f:X \to R^3$ be a smooth curve ...
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0answers
30 views

Definition of Representation in terms of Group Action

The definition of a representation of a group $G$ over a vector space $V$ is a map $p: G \to GL(V)$. According to wikipedia, for finite groups an equivalent definition is an action of $G$ on $V$. ...
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1answer
32 views

Fundamental domains of Dihedral groups

Let $D_n$ be dihedral group of order $2n$, it acts on plane $\mathbb{R}^2$ in a standard way, by rotations and reflections. How one can find fundamental domains for such action?
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2answers
31 views

How do open subsets of X/G look like?

Let $G$ act continuously on $X$, where $X$ is a topological space. So I wonder about how open subsets look like in $X/G$. The action $a$ is defined as $a(g,x)=g.x$.
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23 views

Group action on a tensor product

Let $R \subset S$ be an extension of commutative rings, $G$ a group and $M$ a left $R[G]$- module. Then how do I make the tensor product $S\otimes_R M$ into a left $S[G]$- module? What is the action ...
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2answers
22 views

Definition of equivariant map if one of the actions is a right action

Let $G$ be a group and $f:X\to Y$ a map between two $G$-sets which preserve the $G$ action. If $X$ has a left $G$ action and $Y$ right $G$ action then why do we define $f(g.x)=f(x).g^{-1}$ for all $ ...
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1answer
32 views

Orbit spaces of linear involutions on spheres

Consider the following involutions ($\mathbb{Z}_2$-actions) on the unit $2$-sphere $S^2 \subseteq \mathbb{R}^3$: $(x, y, z) \mapsto (-x,-y,-z)$, the antipodal action; the orbit space is the ...
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1answer
53 views

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I've come up with a proof for the following question, and I'd like to know if it's correct (as I couldn't find anything online along the lines of what I did). Question Let $p$ and $q$ be primes ...
2
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0answers
45 views

Hausdorffness of quotient space

Let $G$ be a compact topological group, and $X$ be a Hausdorff space. We assume that $G$ acts on $X$. Is the quotient space $X/G$ with the quotient topology a Hausdorff space? It seems that the ...
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1answer
48 views

How to think of group actions?

I am a little confused on how exactly I should be thinking of an action on a group. I have been trying to read up on it and came across Timothy Gower's blog which I think does a good job explaining ...
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1answer
35 views

Group actions and permutation representation

Im trying to solve this problem from Dummit & Foote: Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma\in G$ ...
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40 views

Group action of a direct product of groups

Let $G$ be a finite group acting on the $n$-dim vector space $X$. Let $R$ be an $n$-dim representation of $G$. $X$ consists of points $(x1,x2,...,xn)$, which are acted upon by $R(g)$, for $g$ in ...
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1answer
32 views

Index of center $Z(G)$ is finite implies the number of elements of conjugacy class is finite

Exercise Let $G$ be a group such that its center $Z(G)$ has finite index in $G$. Show that every conjugation class has finite elements. I don't know how to attack the problem. I thought the ...
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38 views

Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
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1answer
45 views

Action of Symmetric Group on Lie Polynomials with GAP

Let $L$ be the free Lie Algebra, freely generated by $x_1,x_2, \ldots, x_n$. Let $f$ be a polynomial in $L$ and $\sigma \in S_n$, how to do $\sigma$ act on $f$ in GAP? That is $$\sigma f(x_1, \ldots, ...
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53 views

Why is this action faithful? (Proof that the automorphism group of $A_n$ is $S_n$ for $n\geq 7$.)

I'm currently trying to work through a proof that Aut$(A_n) \cong S_n$. In particular I'm looking at theorem 2.3 (on page 18) in R. Wilson's book 'The Finite Simple Groups'. (Click here for a download ...
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2answers
18 views

How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...
2
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0answers
27 views

Sufficiently transitive implies alternating sans Enormous Theorem

According to this webpage and this mathworld article, if $G<S_n$ is a permutation group which acts sextuply transitively then $G=A_n$ is the alternating group, but this fact is known on the basis ...
4
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1answer
33 views

Product of two stabilizers of transitive group action is proper subset of G?

Suppose $G$ is a finite group and G acts transitively on some set $X$. Let $a$ and $b$ be two distinct elements of $X$ and $G_{a}$ and $G_{b}$ be stabilizers of $a$ and $b$ respectively.Show that ...