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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1
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1answer
23 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
6
votes
1answer
58 views

Number of ways to pick N numbers from 0,1,…,N-1, with possible duplication, with sum equal 0 mod N

We have the numbers $0,1,2,....,N-1$ in $\mathbb Z_N.$ I want to pick $N$ numbers from these. These are the rules: Duplication may occur We don't care about ordering, $00041$ is equivalent to ...
1
vote
1answer
63 views

Group action on $\mathbb R^2$: are my thoughts correct?

Let $G=\mathbb Z / n \mathbb Z$ for $n > 2$ and let $G$ act on $\mathbb R^2$ linearly and effectively. Let $T_g (v)$ denote the element $gv$ where $v \in \mathbb R^2$. Assume that $\det T_g > 0$ ...
3
votes
1answer
31 views

My proof that $G(x)\to G / G_x$ is injective

Please could someone check my proof that $\varphi : G(x) \to G/G_x$ is injective? The notation is the following: $G$ is a group acting on a set, $G_x = \{g \in G\mid gx = x \}$ and $G(x) = \{gx ...
2
votes
1answer
67 views

Group Actions: Orbit Space

Given a group action $G\curvearrowright X$. Consider the orbit space: $\pi:X\to X/G$ Do continuous group actions correspond to open projections, i.e.: $$l_g\in\mathcal{C}(X)\quad(g\in ...
0
votes
1answer
24 views

A topological space with a transitive action.

Let $X$ be a topological space on which a topological group $G$ acts transitively. Given $x\in X$ let $$stab(x)=\{g\in G\;|\; gx=x\}.$$ I want to show that $X$ is homeomorphic to $G/stab(x)$ for any ...
1
vote
1answer
65 views

Group Actions: Discontinuity

Given a group action $G\curvearrowright X$. Then it need not be a continuous one: $l_g\notin\mathcal{C}(X)$ As an example I have in mind: $$k\in\mathbb{Z}:\quad l_k(x\in\mathbb{Z}):=x+k,\quad ...
2
votes
1answer
31 views

Symmetric Group acting on $X \times X$

The symmetric group $S_n$ acts on the set $X = \{1,\ldots,n\}$ and hence acts on $X \times X$ by $g(x,y) = (gx, gy)$. Determine the orbits of $S_n$ on $X \times X$. Not sure how do I actually ...
1
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0answers
23 views

Orbit space of action of a subgroup of a Lie group on a separable metric space

I am stuck on this question. Let $G$ be a Lie group acting freely on a separable metric space $X$. Assume that the orbit space $X/G$ is Hausdorff. Let $H$ be a normal Lie subgroup of $G$. Is the orbit ...
4
votes
3answers
144 views

Group acting on its subsets

Let $ G $ be a group with $ |G|=mp^\alpha $ where $ \alpha\geq1 $ and p is prime integer with $p \nmid m$. Then denote the set of subsets of G, having $p^\alpha$ size, with $X$. Then with the action ...
2
votes
1answer
77 views

Existence of a particular group action

Let $P$ be a group with normal subgroups $G$ and $H$, with $G \not \subset H$, $H \not \subset G$ and $G \cap H \neq 1$. Consider group actions $\theta : G \to Aut(H)$ and $\xi: H \to Aut(G)$ such ...
0
votes
0answers
22 views

Principal orbit type

I have trouble understanding the proof of Proposition 1.2.5 on p.17 in Audin's Torus Actions on Symplectic Manifolds: Let $G\curvearrowright M$ be a smooth action of a compact Lie group $G$ on a ...
1
vote
3answers
49 views

What is the relation between $\mathbb{C}[M]$ and $\mathbb{C}[M/U]$.

Let $M$ be a variety and let $U$ be a group. By definition, $M/U$ is the space of all $U$-orbits of $M$. Now we take coordinate rings $\mathbb{C}[M]$ and $\mathbb{C}[M/U]$. What is the relation ...
2
votes
1answer
36 views

Burnside's lemma simple use

Let's say that $D_3$ acts on a bracelet of 3 beads (Denote S), each bead can be Black or White. I want to count the number of different bracelets (4 - I believe) But using burnside's lemma I get ...
1
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2answers
36 views

Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
2
votes
1answer
36 views

Subgroups and an union of orbits

I have to prove or disprove the following statement: If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an ...
2
votes
1answer
45 views

Size of the orbits of a normal subgroup

So this is the question: Let $H$ be a finite subgroup of $G$, and let $(h,h')(x)=hxh^{-1}$ define an achtion of $H\times H$ on $G$, prove that $H$ is a normal subgroup of $G$ if and only if every ...
0
votes
1answer
21 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 ...
1
vote
1answer
29 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. ...
0
votes
0answers
12 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
votes
0answers
34 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 ...
0
votes
2answers
36 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 ...
3
votes
0answers
72 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
votes
1answer
34 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
votes
1answer
33 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
vote
1answer
45 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
votes
1answer
33 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
votes
0answers
34 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) ...
9
votes
0answers
181 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 ...
0
votes
0answers
16 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
votes
2answers
60 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 $o:G \times X \to X$ ...
0
votes
1answer
17 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 ...
0
votes
0answers
25 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
43 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
vote
1answer
23 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
votes
0answers
28 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 ...
0
votes
0answers
59 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 ...
0
votes
1answer
36 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 ...
1
vote
0answers
19 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
27 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 ...
1
vote
2answers
31 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 ...
0
votes
0answers
20 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 ...
0
votes
2answers
65 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
votes
1answer
33 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
43 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
49 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
votes
2answers
66 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
votes
1answer
56 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
65 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
41 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 ...