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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2
votes
1answer
42 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
1
vote
1answer
29 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
0
votes
1answer
11 views

Is the orbit map for a group action closed in this case?

Suppose a compact Lie group $G$ acts on a manifold $M$ and let $\pi : M \rightarrow M/G$ be the orbit map. Can I say that $\pi$ is closed map? If $C \subseteq M$ is a closed set in $M$ then I only ...
1
vote
0answers
25 views

G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
3
votes
2answers
24 views

Group Actions: Verify a Bijective Correspondence

This is an old exam problem: Given an action of $G$ on $X$, we can define $\varphi: G \to S_X$ by the rule $\varphi(g) = \sigma_g$, where $\sigma_g$ is left multiplication by $g \in G$. Prove that ...
2
votes
2answers
34 views

Does transitive imply it's the entire symmetric group

Let $G$ denote a finite group and recall that $G$ acts transitively (on itself) if and only if for all $x,y \in G$ there is a $g \in G$ such that $gx = y$. I am wondering if transitive may imply that ...
1
vote
3answers
19 views

The Stabilizer of the coset for the action of G on $G/H$ by left multiplication.

Let $H$ be a subgroup of $G$. What is the stabilizer of the coset $aH$ for the action of $G$ on $X=G/H$ by left multiplication? So, I think I've done this one correctly: The Stabilizer is of the ...
0
votes
1answer
21 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
0
votes
0answers
15 views

Continuous action on tensor product

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple ...
2
votes
1answer
12 views

Group orbit computations

Can someone please verify these? Compute the orbit of a vertex of the square under the tautological action of $D_4$. Compute the orbit of a point $x \in G$ under the action of $G$ on itself by ...
1
vote
1answer
17 views

Characterize a faithful action in terms of stabilizers [closed]

Characterize a faithful action in terms of stabilizers. Is the following correct? An action of group $G$ on set $X$ is faithful if and only if $$\bigcap_{x \in X} \mathrm{Stab}_G(x) \neq \{ e ...
2
votes
1answer
44 views

the largest prime number $p$ dividing $ \left|G\right| $ is also divide $ \left|X\right| $.

Let $ G $ be a finite simple group which acts on finite set $ X $ non-trivially. The goal is that the largest prime number $p$ dividing $ \left|G\right| $ is also divide $ \left|X\right| $. I know ...
3
votes
0answers
27 views

Transfer homomorphism in transformation groups

I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups. But suppose ...
0
votes
0answers
12 views

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 ...
1
vote
1answer
29 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 = ...
1
vote
1answer
45 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 ...
-1
votes
0answers
24 views

Isometry Group acts by isometries on space

This is a rather trivial question. But does the isometry group $\text{Isom}(X)$ of a space $X$ act on $X$ by isometries?
0
votes
1answer
26 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 ...
0
votes
1answer
27 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 ...
0
votes
0answers
14 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 $G/S$ ...
1
vote
0answers
46 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 ...
0
votes
0answers
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 ...
2
votes
2answers
73 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 ...
9
votes
0answers
199 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 ...
0
votes
0answers
26 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 $G$ ...
1
vote
1answer
34 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 ...
3
votes
0answers
33 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 ...
2
votes
2answers
70 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 ...
1
vote
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 ...
0
votes
0answers
23 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 ...
0
votes
1answer
24 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 ...
12
votes
0answers
155 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 ...
1
vote
0answers
11 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 $$ ...
0
votes
1answer
46 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 $$ ...
1
vote
0answers
30 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 ...
4
votes
1answer
71 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 ...
2
votes
0answers
43 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 ...
0
votes
0answers
18 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 ...
2
votes
1answer
35 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 ...
1
vote
2answers
41 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 $H=\phi^{-1}(A_{2n})$. ...
4
votes
1answer
65 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 ...
0
votes
0answers
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( ...
0
votes
1answer
28 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 ...
2
votes
1answer
39 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 ...
0
votes
1answer
35 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 ...
3
votes
0answers
35 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 ...
2
votes
3answers
64 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 ...
1
vote
1answer
81 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 ...
1
vote
1answer
37 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 ...
3
votes
1answer
81 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 $x$, ...