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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0
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2answers
39 views

If $X$ and $Y$ are g-equivariant homeomorphic then $X/G$ and $Y/G$ are homeomorphic

Let $X$ and $Y$ be $G$-sets (That is the group $G$ acts on $X$ and $Y$). We say that the function $f: X\to Y$ is G-equivariant if $f(g.x) = g.f(x)$ for all $x\in X$ and all $g\in G$. Prove that if ...
2
votes
2answers
43 views

Group action and Right action

Sorry if this may seem trivial - I just started studying Group Theory. This is the problem: Prove that $(g,h) \rightarrow hg$ does not define a group action with $g$ acting on $h$. Prove instead ...
-1
votes
1answer
42 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
1
vote
2answers
61 views

Intuition behind group action on a set

In Algebra Chapter 0 the definition of a group action on a set is given as: An action of a group $G$ on a set $A$ is a set function $P:G\times A\rightarrow{A}$ such that $P(e_G,a)=a$ and ...
5
votes
3answers
76 views

How many faces, edges and vertices are fixed when $S_4$ permutes the diagonals of a cube?

Consider the action of $S_4$ on a cube, where it acts by permuting the long diagonals. The conjugacy classes of $S_4$ are denoted by $id$, (12), (123), (1234) and (12)(34). I want to know the number ...
1
vote
1answer
70 views

Computing the size of the stabilizers when $U(q)$ acts on $\Bbb Z_q$

Let $\mathbb{Z}_q$ be the additive group of integers modulo $q$ and $U(q):=\{g\in\mathbb{Z}_q:(g,q)=1\}$. If $a\in\mathbb{Z}_q$, then what is the cardinality of the set $\{g\in U(q):ga\equiv a(\mod ...
1
vote
1answer
20 views

Poisson actions defined in terms of coactions.

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, ...
0
votes
1answer
45 views

Action of automorphisms on Eisenstein series

Let $ f \in \mathcal{M}_{k}(\Gamma) $ and $ \sigma \in \textit{Aut}(\mathbb{C}) $. Suppose $$ f = \sum_{n=0}^{\infty}a_{n}q^{n} .$$ Then we define the action of $ \textit{Aut}(\mathbb{C}) $ on $ ...
3
votes
1answer
45 views

Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...
3
votes
1answer
90 views

Isomorphic but not equivalent actions of a group G

This is in some sense a continuation of this problem. Given a group $G$ I would like to exhibit two actions of $G$ on a set $[n] =\{1,\ldots,n\}$ such that the two actions are isomorphic yet not ...
3
votes
1answer
27 views

how to see whether a bundle is trivial or not?

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
3
votes
2answers
73 views

Group action with two orbits

Suppose a group $G$ acts faithfully on a set of five elements, inducing two orbits of size $3$ and $2$ respectively. What group may $G$ be? There is clearly a homomorphism $G \mapsto S_3$ and ...
0
votes
0answers
23 views

equivariant map and cocompact action on subgroup.

Could anybody help me with the following? Let $G,H$ be two hyperbolic groups acting on $X$ and $Y$ respectivly. Let $f:G\rightarrow H$ be map between these groups and let $q:X\rightarrow Y$ be a an ...
0
votes
0answers
29 views

Question on proper group actions

I am trying to prove the following - If $G$ is a discrete topological group acting properly on a Hausdorff space $X$ then the isotropy group (also called stabilizer) $G_x$ of $x \in X$ is finite. ...
3
votes
1answer
56 views

First examples for topology of non-Hausdorff spaces

I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions). As a first example, ...
2
votes
1answer
54 views

Permutation isomorphic subgroups of $S_n$ are conjugate

Consider $G,H \leq S_n$ and their natural action on $[n] = \{1,\ldots,n\}.$ We say that $G$ and $H$ are permutation isomorphic if there is a bijection $\varphi:[n] \mapsto [n]$ and group isomorphism ...
1
vote
1answer
37 views

Describe invariants using coaction.

Let $X$ be an algebraic variety and $G$ be an algebraic group which acts on $X$. We know that an invariant function $f$ in the coordinate ring $\mathbb{C}[X]$ is a function such that $g(f) = f$ for ...
0
votes
1answer
25 views

If $Aut_K(L)$ operates transitive over the zeros of $f$, then $f$ is irreducible

Let $K$ be a field, $f\in K[X]$ a separable polynomial and $L$ a splitting field of $f$. Show that if $Aut_K(L)$ operates transitive over the zeros of f, then f is irreducible. Can someone help?
1
vote
1answer
28 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
72 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
69 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
36 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
131 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
31 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
72 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
35 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
vote
0answers
36 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
172 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
83 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
33 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
50 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
66 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
vote
2answers
44 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
47 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
49 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
23 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. ...
1
vote
0answers
19 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
1answer
44 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
69 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
78 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
44 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
56 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
35 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
35 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) ...
10
votes
1answer
216 views

Writing $G/A\times G/B$ 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
19 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
63 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
25 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 ...