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
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2answers
117 views

Diagonal groups and semidirect products

I am studying a text on permutation groups, which has the following example in a section on regular normal subgroups: If $Z(N)=1$, then $N \cong \mathrm{Inn}(N)$, the group of inner ...
2
votes
3answers
46 views

Prove that $ \pi_g $ is a permutation

Suppose that $ G $ is group and $ g $ is any element from $ G $ ($g \in G $). We define a function $ \pi_g : G \rightarrow G,\ \ \pi_g(x) = g \cdot x$. Prove that $ \pi_g $ is a permutation of the set ...
3
votes
1answer
118 views

A group action contains another action as a normal subgroup

I have a question on what it means for one group action to contain another group action as a normal subgroup. I am guessing this means that the image of the first group action contains the image of ...
3
votes
1answer
168 views

Left regular action isomorphic to the right regular action

I am reading something on regular groups and I have a question on why the left and right regular actions are isomorphic. Let $G$ be a group. Consider the homomorphism $\rho: G \rightarrow Sym(G), g ...
1
vote
1answer
297 views

Number of Orbits in Group Action

Let G be a group of order 15 acting on a set of order 22. Assume there are no fixed points. Determine how many orbits there are.
1
vote
2answers
106 views

$G$-set terminology

When a group action $G \times X \rightarrow X$ is defined with a group $G$ and a set $X$, why is there not a special name for the set $X$? I know that this is referred to as a $G$-set, but the set $X$ ...
3
votes
2answers
66 views

action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
0
votes
0answers
45 views

boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on ...
0
votes
0answers
44 views

Interpreting Notation (stabilizer)

Let $f$ be a separable polynomial, $H, \subset L \subsetneq S_n$ groups, $L/H=\{ H,...,t_e H \}$, $\Theta \in k[x_1,...,x_n]$ s.t. $\mathrm{Stab}_L(\Theta)=H$, $\theta=\widetilde{\Theta}=$evaluation ...
11
votes
1answer
266 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
1
vote
0answers
65 views

Tangent bundle of a quotient manifold

I am interested in the tangent bundle of the quotient of a manifold by a proper and free action but I can't find any reference on the net. Does anyone know a book or article where it is described ?
1
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0answers
42 views

When the $(\mathbb R,+)$-action defined by the flow of a vector field on a manifold is proper ?

I'm interested in the conditions that a vector field has to satisfy in order for its flow to define a proper action. Technically, let $\xi$ be a smooth vector field. For each $m\in M$, there is a ...
0
votes
1answer
111 views

Kernel of this action is trivial?

Let $S_n$ act on $S_n/H=\{ H,t_2 H,...,t_e H\}$ from the left and let $n\geq 5$, $e \geq 3$. Then the kernel of this action is trivial. This supposedly follows because $e \geq 3$, but I haven't the ...
1
vote
0answers
44 views

Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
3
votes
1answer
112 views

Help with Conrad's “Recognizing Galois Groups”

I'm trying to do the proofs of theorems 2.1 and 2.2 in this article by Keith Conrad. I'll just quote the parts of theorem 2.1 I have problems with: if $a \sim b$ then $ga \sim gb$... if $a \neq b$ ...
2
votes
1answer
55 views

fundamental group of a graph is free

Let $X$ be a connected graph, and $T$ its maximal tree. Via covering spaces and deck-transformations, how one can prove that $\pi_1(X)= \pi_1(X/T)$?
4
votes
0answers
71 views

Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
2
votes
1answer
85 views

Splitting field of resolvent equals that of $f$

Lemma: Let $\Psi \in k[X_1,...,X_n]=:B$ be s.t. $stab_{S_n}(\Psi)=H \subset S_n$, $S_n/H=\{ \Psi, t_2 \Psi,..., t_e\Psi \}$, $\Delta_\Psi$ the discriminant of $L_\Psi:=\prod_{i=1}^e (X-t_i \Psi)$, and ...
1
vote
1answer
51 views

Partitions and Orbit Sizes

If $U,V \subset S_n$ are subgroups with $S_n//U = \{id,g_2,...,g_e\}$ and $\alpha_j$ is $\frac{1}{j}$ times the number of $i\in [e]$ s.t $[V:V \cap g_i U g_i^{-1}]=j$ then $(\alpha_1,...,\alpha_e)$ is ...
1
vote
1answer
75 views

Show that the stabilizer is a prime subgroup

We define a subgroup $H$ as being convex if $g\in H\implies h\in H$ for all $1\leq h\leq g$. A convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that ...
1
vote
0answers
54 views

Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
0
votes
1answer
34 views

Quotient by the action of positive reals

Today in my complex analysis class Riemann sphere was defined, and of course the construction caused questions, such as "why don't we distinguish between all the various infinities?" and "Would it ...
2
votes
0answers
101 views

Degree of factor in a resolvent

Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma ...
4
votes
2answers
83 views

Faithful group actions and dimensions

Just a quick question. I'm trying to understand the answer to one of my previous questions. The precise problem I want to show is as follows. Let $G$ be a group acting faithfully on a manifold ...
2
votes
0answers
89 views

trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...
4
votes
0answers
110 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
5
votes
0answers
102 views

Group action on a manifold with finitely many orbits

I'm looking for a result along the lines of the following: Let $G$ be a group acting on a set $X$. If the action partitions $X$ into finitely many $G$-orbits, then $\dim G \geq \dim X$. For ...
6
votes
1answer
394 views

Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
2
votes
1answer
66 views

Are subgroups kernels of some $1$-cocycles?

If an $1$-cocycle $\sigma :G\to K$ has image in a group on which $G$ acts, then the set of element of $G$ that is mapped to $\text{$e_K$}$ by $\sigma$ forms a subgroup of $G$. This could be proved by ...
2
votes
1answer
66 views

Group acting by isometries on a length space

I am reading the book A course in metric geometry by Burago, Burago and Ivanov. I have some difficulties with an exercise 3.4.6 on page 78. The exercise is the following: Let a group $G$ act by ...
2
votes
2answers
100 views

Proving those elliptic matrices in $\operatorname{SL}_2(ℤ)$ are not conjugate

Set $\mathbf{\Gamma} = \operatorname{SL}_2(ℤ)$, let $\mathbf{H}$ denote the upper half plane. and let $$\Gamma_0 (N) = \left\{ \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] ...
1
vote
1answer
36 views

What subsets of a covering space cover their image?

Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map? If it makes things easier, assume $E$ is simply connected, that is, the ...
7
votes
3answers
141 views

Realizing groups as symmetry groups

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ...
2
votes
1answer
74 views

How to find the order of $ X_k$?

Let $G_k = \Bbb Z_3 × · · · × \Bbb Z_3$. Let$ \,\,\alpha(z_1, . . . , z_{k−1}, z_k )=(−z_1, . . . ,−z_{k−1}, z_k ) \text{ where} \,\,z_i \in\Bbb Z_3$ for $i = 1, 2, . . . ,k$. Then $α ∈ ...
15
votes
0answers
150 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
4
votes
2answers
171 views

Group Actions of $S_n$ and $O(n)$

I have been reading up a bit on group actions and whether they are faithful, free and transitive. A question I found states: Consider the following group actions 1) The symmetric group $S_n$ ...
-1
votes
1answer
71 views

Group acting on a set

Let $A$ be a set, and let $G$ be any subgroup of $S_A$. $G$ is a group of permutations of $A$; we say it is a group acting on the set $A$. Assume here that $G$ is a finite group. If $u \in A$, the ...
2
votes
0answers
71 views

An orbit of a group action and the implicit function theorem

Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
3
votes
1answer
45 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
196 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
114 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
82 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 $\alpha$ ...
4
votes
0answers
171 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
51 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) ...
3
votes
2answers
131 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
80 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
122 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
122 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
92 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
47 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 ...