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
66 views
In a transitive action there is a bijection between the fixed points of a stabilizer of a and the lateral clases of the stabilizer in his normalizer
the question is the given above, specially in the case infinite:
If the action of $G$ is transitive, then there is a bijection between the fixed points of the stabilizer of a element $a$ and the ...
3
votes
1answer
34 views
Infinite imprimitive non abelian group?
My new question is
Is there an infinite, imprimitive and non abelian group?
Thank you for the further answers.
2
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1answer
51 views
Function spaces and transitive group actions
Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first.
Let $B$ be a topological space and $G$ a topological group ...
2
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0answers
84 views
Group actions and associated bundles
Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
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1answer
61 views
Let $G$ be transitive.Then $\beta\in \operatorname{fix}(G_\alpha)$ implies $G_\alpha = G_\beta$
i am new in this forum.
My question is about group actions
We have a transitive action of $G$ and $\beta$ a element in the fixed points of the stabilizer of another element $\alpha$. Then $\alpha$ ...
8
votes
3answers
203 views
What does an outer automorphism look like?
I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...
1
vote
0answers
31 views
Action of a Lie group on a coset of its subgroup
I am a physicist, so sorry for the lack of rigor. It is well known that a (say compact) Lie group $G$ acts naturally by left multiplication on the coset space $G/H$ where $H\subset G$ is its (Lie) ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
2
votes
1answer
82 views
In a group of Möbius transformations, does discontinuity imply discreteness?
Let $G$ be a subgroup of the group of Möbius transformations
$$ z \mapsto \frac{az+b}{cz+d}.$$
What is the relationship between the two conditions:
(1) $G$ being discrete.
(2) $G$ acting properly ...
4
votes
1answer
165 views
Quotient of a locally compact Hausdorff space by a proper action is Hausdorff
I am trying to prove the following:
Let $G$ be a topological group acting properly on a Hausdorff locally
compact space $X$, i.e. preimages of compacts sets by the map
$$G\times X\to X\times ...
2
votes
0answers
39 views
Semi-orbital equivalence relation
Edit: I was in kind of a hurry when writing this post and made a mistake in the formula defining $G_E$. What I had written said that $G_E$ preserves the set of classes of $E$, while I meant actually ...
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0answers
72 views
Show that $h \equiv 1 \pmod p$, where $h$ is the number of subgroups of order $p$ and $p$ divides the group order. [duplicate]
Let $G$ be a finite group and $p$ a prime number that divides the
order of $G$. Let $h$ be the number of subgroups of $G$ of order $p$.
Prove that there are $h(p-1)$ elements of order $p$ in ...
4
votes
2answers
71 views
Free objects in $\mathrm{Set}(G).$
What are the free objects in the category of $G$-sets for a group $G$?
After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
2
votes
2answers
109 views
Parabolic isometries on Gromov hyperbolic spaces
Let $X$ be a $\delta$-hyperbolic geodesic space. Then we have the following classification of isometries on $X$:
Theorem: Let $g$ be an isometry on $X$. Then, exactely one of the following case ...
4
votes
2answers
70 views
About the category $\mathrm{Set}(G)$
I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
4
votes
0answers
62 views
The classifying space of a gauge group
Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
3
votes
1answer
64 views
The order of a conjugacy class is bounded by the index of the center
If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)
Here is my attempt: we have
...
3
votes
1answer
57 views
Is the Structure Group of a Fibre Bundle Well-Defined?
Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
5
votes
1answer
74 views
A question about quotient under group action
Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ?
If so, can you please ...
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votes
1answer
25 views
Need to show that order of orbits under group action is non-trivial and intersection of two p-groups is a proper subgroup
I'm working my way through the second and third sylow theorems in my book. Here's the relevant bit:
We have a group $G$ of order $p^\alpha m$ where $p$ does not divide $m$. We have that $Q$ is a ...
1
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1answer
27 views
Question about Sylow's Theorem/Conjugation of the set of conjugates of P
I'm trying to understand a proof of the 2nd and 3rd parts of Sylow's Theorem. In some preliminary work, my book establishes that $P$ is a Sylow p-subgroup of $G$.
Then it defines ...
10
votes
2answers
100 views
What's the idea of an action of a group?
I know the formal definition of an action over a set. I'm not asking this.
What I'm asking is: what's the intuition of it? It is a way to define an algebra over a set? Since an action can exist in ...
1
vote
1answer
32 views
Studying the action of $GL(V)$ on the vector space $V$
The statement I am trying to prove is the following.
Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let
$\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural
...
1
vote
2answers
50 views
Different actions of an affine primitive group?
Fairly new to group actions and I'm having trouble finding answers to these in textbooks...
Say we have a primitive action of $G$ on $\Omega$, with regular elementary abelian socle $N$. Now suppose ...
1
vote
0answers
27 views
Orbits of the action of G/H
Let $G \subset Iso(M)$ be a Lie group which acts on a (semiriemannian) manifold $M$ properly and smoothly. Let we know the orbits of the action. Suppose that $H$ is a discrete central subgroup of $G$ ...
1
vote
2answers
170 views
Rotational Symmetries of a Cube
Use the Orbit Stabilizer Theorem to deduce the number of elements in the rotational symmetry group of the cube.
I can write $\operatorname{Stab}_G(v) = \left\{g \in G \mid g \cdot v = v\right\}$ and ...
3
votes
1answer
25 views
Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types.
Show that the orbits of $S_n$ under the conjugation action of $S_n$
on itself correspond 1-1 with the cycle types.
So, the orbit of $\sigma \in S_n$ is the set $S_n \sigma = \{ \tau .\sigma : ...
3
votes
1answer
55 views
Conjugation on subgroups of $A_4$ faithful?
Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$
I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
4
votes
0answers
42 views
Equiv Relation of Orbits - Group Action [duplicate]
Let $G$ be a group that acts on $X$. I want to show that the orbits of $G$ partition $X$. I am given the relation $x\sim y \iff x\in Orb(y)$. Now:
$x\sim y\iff x\in Orb(y) \iff x=gy$ for some $g\in G ...
2
votes
2answers
37 views
Homsets of group actions related to fixed points
MacLane and Moerdijk's Sheaves in Geometry and Logic has a section on Continuous Group Actions (Sec. III.9). On page 152, there is an isomorphism displayed:
$$Hom_G(G/U, X) \cong X^U$$
In their ...
2
votes
2answers
241 views
Complex projective line hausdorff as quotient space
I was wondering if there is a simple argument showing that the complex projective line defined as $\mathbb{CP^1} = \big(\mathbb{C}^2 \setminus \{0\}\big)/{\mathbb{C}^{\times}}$ is hausdorff when ...
1
vote
2answers
92 views
Properties of set $\mathrm {orb} (x)$
Properties of set $\mathrm {orb} (x)$:
${\displaystyle \bigcup_{x\in X}\mathrm{orb}(x)=X}$;
$\mathrm{orb}(x)\cap\mathrm{orb}(y)=\emptyset$
for all $x,y\in X, x\neq y$
How to prove it? Please ...
3
votes
3answers
110 views
Examples of the dihedral group $D_4$ acting on sets
Consider the group $D_4$. Give examples of $D_4$ acting on a set.
Attempt: So $|D_4| = 8$. I have come up with a few, but I was wondering what some people here thought.
First one we came up with ...
2
votes
2answers
70 views
How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$?
Suppose the additive group $\mathbb Z^n$ acts on $\mathbb R^n$ through translation. How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$? The translation action is given by ...
0
votes
1answer
23 views
Prove this action is properly discontinuous..
Consider the group $\mathbb Z_2=\{0, 1\}$ acting on the sphere $\mathbb S^n$ through the group actions $\psi_0=Id$ e $\psi_1=-Id$. Show this actions is properly discontinuos? The definition of ...
1
vote
1answer
27 views
How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$?
How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$? Here $\mathbb Z_2=\{0, 1\}$ is the additive group and the group action considered induces the aplications $\psi_0=Id$ and ...
1
vote
1answer
42 views
right group action
wikipedia says 'The difference between left and right actions is in the order in which a product like $gh$ acts on $x$. For a left action $h$ acts first and is followed by $g$, while for a right ...
4
votes
3answers
53 views
at least one element fixed by all the group
$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$?
I tried to ask this question ...
2
votes
0answers
72 views
Group action and Radon measure
Let $\mathscr M(\mathbb R)$ be the Banach space of complex-valued Radon measures on $\mathbb R$, and let $\pi$ be the action of $\mathbb R$ on $\mathscr M(\mathbb R)$. Let $\mathscr A$ denote a subset ...
2
votes
1answer
37 views
Prove: Let $Gal(f)$ acts transitively on $Z(f)$ if and only if $f$ is irreducible in $F[x]$
Can someone provide a proof for this, please? Particularly for the backward direction.
Let $F$ be a field.
Let $f(x)$ be a separable polynomial in $F[x]$.
Let $K/F$ be the splitting field of $f(x)$.
...
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votes
2answers
55 views
Group and orbit question.
Suppose group $G$ acts on a set $A$.
a) If $x$ and $y$ are in the same orbit, show that there exists some $g \in G$ such that $gG_x g^{-1} = G_y$.
b) Show that if $|G.x|$ is finite, then $|G.x| = ...
0
votes
1answer
80 views
p-group and group actions
$G$ is a $p$-group, which means $|G|=p^n$ for $n\in \mathbb{Z^+}$.
Now,if $p$ does not divide $|S|$, for S is a set that G acts upon, how do I show that there exists $a\in S$ such that $G_a=G$
So ...
3
votes
0answers
36 views
burnside lemma cube [duplicate]
Having n colors, use the lemma to find a formula for the number of ways to color the edges of the cube.
What I have so far:
I got $|A/G| = \dfrac{n^{12} + 6n^3 + 3n^6 + 8n^4 + 6n^7}{24}$ but when I ...
2
votes
1answer
110 views
Using Burnside's lemma on the cube.
Having $n$ colors, use the lemma to find a formula for the number of ways to color the edges of the cube.
Here is what I have so far: The Burnside lemma says that $\displaystyle |X/G| = ...
6
votes
1answer
77 views
Natural way to define a free action of a finite abelian group
Let $G$ be a finite abelian group. Then $G \simeq \mathbb{Z}_{u_1} \oplus \cdots \oplus \mathbb{Z}_{u_m}$, where $u_{i}$ is a power of some prime number. Without loss of generality I will consider $G ...
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votes
0answers
42 views
Definition of g-orbit of a set
Let $g$ be a Lie algebra and $M$ a manifold, what does mean $g$-orbit of $M$?
2
votes
1answer
57 views
What does “lifted action” mean?
I read about angular moment and linear moment but I don't know what "lifted action" means. Can you explain please?
Thanks. :)
0
votes
1answer
68 views
Orbit and stabilizer question.
Let $K$ be a field. Consider the action of the multiplicative group $K^* := K-\{0 \}$ on the vector space $K^n$ given by scaling.
a) Describe the orbits of this action.
b) Describe the stabilizer ...
0
votes
0answers
25 views
Finiteness of fixed points of a Lie group action
Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$.
Under which assumptions there will be a ...
5
votes
1answer
48 views
How to check the strong ergodicity of the $SL_2(\mathbb{Z})$-action on the torus?
Suppose $\Gamma\subset SL_2(\mathbb{Z})$ is a non-amenable subgroup, especially, $\Gamma=SL_2(\mathbb{Z})$. Consider the natural action of $\Gamma$ on $S^1\times S^1=T^2$.
How to check that this ...
