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.
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.
0
votes
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 ...
2
votes
1answer
50 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 ...
0
votes
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 $\alpha$ ...
2
votes
0answers
81 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 ...
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 $x \in X$ ...
10
votes
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 ...
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 ...
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 ...
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) = ...
0
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
vote
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$ ...
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 : ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $f(x)$.
...
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 ...
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| = ...
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 ...
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 ...
0
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
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
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 ...
6
votes
1answer
103 views
Group actions transitive on certain subsets
Let $G$ be a group acting on a finite set $X$. This also gives an action of $G$ on the subsets of $X$ of any given size, and we can ask whether this action is transitive for some specified size of ...
4
votes
1answer
98 views
wiki's definition of “strongly continuous group action” wrong?
Wikipedia defines strongly continuous group action as follows:
A group action of a topological group G on a topological space X is said to be
strongly continuous if for all x in X, the map g ↦ ...
2
votes
1answer
54 views
Complexifying a group action of SL(n, R) to a group action of SL(n, C)
Given an analytic group action of $SL(n, \mathbb{R})$ on $\mathbb{R}^m$ fixing the origin, in this article the author then proceeds to "complexify the analytic $SL(n, \mathbb{R})$ action to obtain a ...
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 ...
1
vote
1answer
29 views
how to convert right group action to left group action?
In Wikipedia it says one can convert right group action to left group action, because of the formula $(gh)^{−1} = h^{−1}g^{−1}$.
Can you explain how this works?
-3
votes
2answers
127 views
On Conjugacy Classes and Alternating Group $A_n$
in Dummit & Foote
in page 131
"
Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ .
1.Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any odd ...
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 ...
2
votes
0answers
47 views
Infinitesimal generators of actions
Is there a method to obtain an action of an infinite dimensional Lie group starting with its infinitesimal generator ?
I'm interested about actions of G on itself . And I was wondering if I can ...
4
votes
3answers
84 views
Do we gain anything interesting if the stabilizer subgroup of a point is normal?
Let $G$ be a group and $S$ a $G$-set with action $(g,s) \mapsto gs$. For some $s \in S$, let the stabilizer of $s$, $G_s=\{g \in G\,|\,gs=s\}$ be normal in $G$. What does this let us say about the ...
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
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 ...
