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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3
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2answers
114 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
78 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 ...
5
votes
2answers
113 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
113 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
86 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
46 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
53 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
146 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
45 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
65 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
1answer
59 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
98 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
107 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
84 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
44 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
64 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
43 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 ...
7
votes
1answer
165 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
100 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
42 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
221 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 ...
3
votes
1answer
76 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
66 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
87 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
40 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
365 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
110 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
87 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 ...
2
votes
1answer
127 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
101 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
119 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
174 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 ...
5
votes
1answer
308 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
90 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 ...
1
vote
0answers
38 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
50 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?
0
votes
2answers
446 views

On Conjugacy Classes of Alternating Group $A_n$

In Dummit & Foote, page 131 Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ . Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any ...
3
votes
2answers
217 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
115 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
347 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
775 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 ...
2
votes
3answers
208 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 ...
0
votes
0answers
105 views

What is the Lobachevsky space?

I am reading about Lie group actions on manifolds and the author used the Lobachevsky space $SO^{+}_{1,n}/SO_n$ with the actions of the subgroups $SO_n$, $SO_{1,n-1}$ and the horispherical group ...
1
vote
0answers
72 views

Actions of G on corresponding orbits are equivalent for stable maps

Given actions of G on X and on Y, these actions are equivalent if and only if there is a bijection from $X $ \ $ G \rightarrow Y$ \ $G$ so that actions of G on corresponding orbits are equivalent. ...
1
vote
2answers
93 views

Finding the number of orbits

How many orbits are there of $(12)(25)$ in $S_{5}$? Considering the permutation $(12)$, it has $4$ orbits and is as follows: $\{\{1,2\},\{3\},\{4\},\{5\}\}$ and (25) also has 4 orbits and is also ...
6
votes
0answers
39 views

Chern classes of free quotient manoflds

Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
5
votes
1answer
70 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 ...
3
votes
1answer
158 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 ...
1
vote
2answers
57 views

$|G|=11$ operates on $\mathbb{Z} / 5\mathbb{Z}\times \mathbb{Z} / 5\mathbb{Z}$ implies at least one point fixpoint

Can someone please help me with the following: A group of order $11$ operates on $\mathbb{Z} / 5\mathbb{Z}\times \mathbb{Z} / 5\mathbb{Z}$. I have to show that it has at least one fix point. Can ...
1
vote
2answers
120 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 ...