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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1answer
10 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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0answers
17 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
8 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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0answers
8 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 ...
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2answers
29 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 ...
1
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1answer
46 views

In how many ways can the group $\mathbb Z_5$ act on the set $\{1,2,3,4,5\}$? [closed]

In how many ways can the group $\mathbb Z_5$ act on the set $\{1,2,3,4,5\}$? A.5 B.24 C.25 D.120
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0answers
14 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
28 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 ...
0
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1answer
32 views

Conjugacy classes of $S_n$ under the action of $S_{n-1}$

I try to get explicitly сonjugacy classes of $S_n$ under the action of $S_{n-1}$. I believe that in the description of the classes present cycle type of a permutation and yet another parameter. But I ...
1
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1answer
42 views

A simple finite group $G$ with $n$ p-Sylows is isomorphic to a subgroup of $\mathbb A_n$

I am trying to solve this problem: Let $G$ be a finite and simple group, and let $p$ be a prime number such that $p$ divides $|G|$. If $n_p(G)=n$ for $n>1$ (n_p denotes the number of p-Sylows) ...
2
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1answer
30 views

Calculate the number of conjugacy classes of $G$ with $|G| = p^4$ with $|Z(G)|=p^2$

Let $p$ be a prime and $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$. I couldn't think of much except for, if $G$ acts on itself by ...
2
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0answers
31 views

Action on a group descends to an action on its factor group

Let $A$ and $B$ be groups and $N\unlhd A$ is a normal subgroup of $A$. Suppose that $B$ acts on $A$; that is, there exists a group homomorphism (not necessarily monomorphism) ...
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171 views

Writing this $G$-set explicitly as union of orbits

Let $G$ be a finite abelian group, and let $A$ and $B$ be subgroups. I'm interested in $G/A\times G/B$ with its natural $G$-set structure. In $G/A\times G/B$, the stabilizer of any element is $A\cap ...
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0answers
15 views

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 ...
3
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2answers
56 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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votes
1answer
14 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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0answers
23 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, ...
3
votes
1answer
34 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 ...
1
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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 ...
2
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0answers
26 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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0answers
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
34 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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0answers
18 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 ...
1
vote
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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2answers
27 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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0answers
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
60 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 ...
3
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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 ...
0
votes
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 ...
1
vote
1answer
45 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 ...
2
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2answers
61 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
55 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\ ...
1
vote
1answer
62 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 ...
0
votes
1answer
40 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 ...
0
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1answer
20 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 ...
5
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2answers
82 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 $\operatorname{SL_2}(\mathbb Z)$ ...
2
votes
1answer
29 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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0answers
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 ...
1
vote
1answer
56 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 ...
3
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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
31 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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0answers
32 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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0answers
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 ...
4
votes
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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0answers
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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0answers
26 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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vote
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 $ ...