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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In which condition the restriction of the orbit map is open?

Let $(X,G)$ be a group action, must the restriction of the orbit map be an open mapping? In other words, $\phi: G \times X \to X$ is a group action, for any $x\in X$, $\phi_x:G \to Gx$ is an open ...
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31 views

The topology on $X / G$ where $G$ acts on $X$

The elements are orbits, but how do we find the neighbourhoods? In particular, let $G= ( \mathbb R^+ , \cdot )$ and $X=[0, \infty )$. Let $G$ act on $X$ via the usual multiplication. Then $X/G = ...
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1answer
51 views

Group action on finite set of integers: is it necessarily a permutation group?

I'm just starting to learn about group actions and there is something not clear to me. Let $S = \{1, \dots, n\}$ and let $G$ be a group acting on $S$. Does this imply that $G$ is a permutation ...
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1answer
29 views

Action on finite non-abelian group

Let $G$ be a finite, non-abelian group. Show that if $Aut(G)$ acts on $G$ by $\sigma.g=\sigma(g)$ for each $\sigma \in Aut(G)$, $g \in G$, then there exist at least three orbits. I think I could ...
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1answer
33 views

restriction of the orbit map must be open?

Let $(X,G)$ be a group action, must the restriction of the orbit map be an open mapping? In other words, $\phi: G \times X \to X$ is a group action, for any $x\in X$, $\phi_x:G \to Gx$ is an open ...
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16 views

three questions about syndetic sets

$G$ is a topological group. Definition: A subset $S$ of $G$ is said to be syndetic in $G$ provided that $G=SK$ for some compact subset $K$ of $G$. 1.If $S$ is a syndetic subgroup in $G$, then $G/S$ ...
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47 views

Certain principle bundle structure on $\mathbb{R}^{n}\setminus \{0\}$

Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a principle fibre bundle. Here $\mathbb{H}^{2}$ is the Poincare upper plane ...
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19 views

$ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $is a principal $ O(n-k) $-bundle.

I'm trying to prove that $ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $; $ A \longmapsto (Ae_1, ... ,Ae_k) $ (the projection from the orthogonal group to the Stiefel manifold) is a ...
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2answers
74 views

Subgroups as Stabilizers

In any group action, stabilizers are subgroups of the group. Question: Given a finite group $G$, does there always exists a set $X$ and an action of $G$ on $X$ such that every subgroup of $G$ is ...
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268 views

Distributing groups of objects into boxes

How can I enumerate the number of ways of distributing distinct groups of identical objects (but various cardinality) into $k$ boxes such that at most one box is empty $(1)$ and no combination of ...
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34 views

Space of $G$-invariant Riemannian metrics contractible?

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ ...
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1answer
37 views

Determining the orbit decomposition of a certain group action on $R^2$

I have the group $$G = \left\{ \begin{bmatrix}a&b\\0&1\end{bmatrix} : a,b \in R, a \neq 0\right\}$$ and the map $\wedge: G \times R^2 \to R^2$ defined by ...
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43 views

Homogeneous metric on a homogeneous space $G/K$ - is this the same as a $G$ - invariant metric?

I have trouble putting down the notion of a homogeneous Riemannian metric. Suppose we are given a Riemannian manifold $(M,g)$ on which a compact Lie group $G$ acts transitively by isometries (this ...
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2answers
80 views

Finite groups whose non-trivial elements have no fixed points

(I first asked this question on MathOverflow, but was recommended to ask here at Mathstackexchange instead.) I am interested in finite groups $G$ acting on a finite set $X$ with the following ...
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1answer
20 views

Identification of the Lie algebra of an isotropy group with the tangent space - stuck with a statement

I think I am stuck with the following statement that I read on the Encyclopedia of Mathematics website regarding Isotropy representations: "If $G$ is a Lie group acting smoothly and transitively on ...
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24 views

Finite groups whose non-trivial elements have no fixed points [duplicate]

I am interested in finite groups $G$ acting on a finite set $X$ with the following property: (*) fix(g)=$\emptyset$ for all $g\in G\setminus\{1\}$, where fix(g):=$\{x\in X|gx=x\}$ denotes the set ...
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1answer
25 views

The action of the orthogonal group $O_n(\mathbb{R})$ on the Stiefel manifold $V_{k,n}(\mathbb{R}) $ .

I'm trying to prove that $O_n(\mathbb{R}) / O_{(n-k)}(\mathbb{R}) \cong V_{k,n}(\mathbb{R})$ where $ O_n(\mathbb{R}) = \left\lbrace A \in M_n(\mathbb{R}) / A A^t = I_{n}\right\rbrace $ is the ...
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178 views

Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
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13 views

Showing the action of $SO(p,q)$ in the punctured cone of isotropic vectors is transitive

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
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1answer
47 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
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31 views

Deriving that center is subset of centralizer(using group actions only)

So if we consider our group $N_G(A)$ and let the group act on the set A $\in$ P(G) via conjugation and consider the kernel we will get precisely the kernel being the centralizer, and since the kernel ...
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1answer
75 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
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52 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
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19 views

Action of $SL_2(p)$ on intgers mod p by Möbius transformations

This is simple and may have been asked before, but I couldn't find it. I have been asked to 'Define the action of $SL_2(p)$ (the group of 2 by 2 matrices of determinant 1 with entries in ...
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1answer
36 views

Image of homomorphism that acts transitively and contains the stabilizer is surjective.

Let $\phi:G_1\rightarrow G_2$ be a homomorphism of groups. Let $G_2$ act on a set $X$, and let $\phi(G_1)$ act transitively on the same set $X$. Finally, let $p\in X$ be arbitrary, and suppose that ...
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3answers
53 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
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1answer
69 views

Quotient space of $\Bbb C^n$ obtained by action of $S_n$

Consider the action of $S_n$ on $\mathbb{C^n}$ given by: $$\sigma(x_1, x_2, \cdots,x_n) = (x_{\sigma(1)}, x_{\sigma(2)}, \cdots,x_{\sigma(n)}).$$ What is the quotient space of $\mathbb{C^n}$ obtained ...
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60 views

Is this an action of $S_{n}$ on $\mathbb{R}_{n}$?

I am trying to prove that $S_{n}$ acts on $\mathbb{R}_{n}$ with the map $$* : S_{n} \rightarrow \mathbb{R}_{n}, \quad * \left( \sigma, \left( r_{1}, r_{2}, \dots, r_{n} \right) \right) = \left( ...
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1answer
30 views

If $G$ acts non-trivially on a set then it has a proper normal subgroup of finite index

The following is an old exam question from a n introduction to group theory course: Let $G$ be a group (possibly infinite) that acts non-trivially on a finite set $X$ - that is there are $g\in ...
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1answer
44 views

$p$-group acting on a finite set

Let $G$ be a $p$-group. Prove that if $G$ acts on a finite set $X$ and $p$ does not divide $|X|$, then $X$ contains some element that is fixed by every element in $G$. Any thoughts? I'm stumped ...
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1answer
40 views

The stabiliser of $g.x$ is the subgroup $gGxg^{-1}$

Let $X$ a G-set and $x \in X$. Show that for any $g\in G$, the stabiliser of $g.x$ is the subgroup $gGxg^{-1}$. I found this question in the book A Course in Group Theory of J. Humphreys. I was ...
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0answers
36 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
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3answers
81 views

Examples of Non-Faithful Group Actions

I cannot find anywhere a relatively simple example of a non-faithful group action. I feel I understand the definition relatively well, however I can't come up with any ideas for one in my head (and ...
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1answer
95 views

Consider group G acting on a set X

Consider group G acting on a set X Give examples of: a)The action that is transitive and faithful My Answer: Group G under addition acting on a set of integers Z b)The action that is transitive ...
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1answer
41 views

Is the saturation of Borel sets Borel?

Problem. Let $G\times X\rightarrow X$ be a continuous action of a Polish group on a Polish space. Let $A\subseteq X$ be Borel. Is the saturation $[A]_{G}:=G\cdot A$ a Borel set? One approach. The ...
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1answer
85 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
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0answers
65 views

Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
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52 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
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97 views

is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
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1answer
37 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
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2answers
169 views

What is a “natural group action”?

Eg. The symmetric group on S acts on S in a natural way, for all sets S. Thanks in advance!
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1answer
55 views

Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states: Let $G$ act ...
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1answer
67 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
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0answers
36 views

for which values of k is the action of $S_A$ on k-element subsets faithful

I am solving dummit and foote question and I saw this question in one of the chapters. So in order to be faithful we have to have the given action induce distinct permutation on the set, but the ...
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0answers
19 views

cocompact action + finite stabilizers => proper action?

Assume a disctere countable group G acts on a smooth manifold M by diffeomorfisms and (1) M/G is compact and Hausdorff (2) all stabilizers are finite How to prove that the action is proper?
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1answer
30 views

Orbit-Stabiliser Theorem Applied

"The group S6 acts on the group Z6 via σ([a]) = [σ(a)], for σ ∈ S6 and a∈{1,...,6}. A permutation that is also an isomorphism is called an automorphism. The set G of automorphisms of Z6 is a group. ...
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3answers
155 views

What is a group action, and how can we apply it to Sylow theory

I am studying Sylow theorems at the moment, more specifically trying to solve the following problem that I recently posted: Let G be a finite group which has exactly eight Sylow 7 subgroups. Show ...
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1answer
129 views

If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. Now this is my ...
2
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1answer
161 views

Defining action of an elementary abelian 2-group on a vector space.

I have a group $G = \oplus_{\alpha} \mathbb{Z}/2$, where all the direct summands are indexed by the elements of a set (a list in GAP). I want to define in GAP an action of this group on a vector ...
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1answer
44 views

Proving that the dihedral group $D_n$ has $2n$ elements

I am trying to prove that the dihedral group $D_n$ has $2n$ elements by using the theory of group actions. Specifically I want to use the orbit stabilizer theorem. So I need $D_n$ to act on a specific ...