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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27 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 ...
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0answers
60 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.
...
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2answers
66 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
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0answers
34 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
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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 ...
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
...
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2answers
54 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 ...
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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 ...
1
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1answer
34 views
According to my solution there should be more fixed points…
I have solved the following exercise:
A group of order $55$ acts on a set of order $18$. Then there are at least $2$ fixed points.
But according to my solution, there should be at least 3 fixed ...
1
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1answer
85 views
Stabilizer of a 4 by 4 skew symmetric matrix by orthogonal matrix
Matrices are over the field of complex numbers, and $X^t$ means transpose of a matrix $X$.
Consider the group action of $O(4)=\{P\mid PP^t=I\}$ on $SK(4)=\{M\mid M^t=-M\}$ by $(P,M) \rightarrow ...
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1answer
114 views
On Group action and blocks of subgroups of the symmetric Group
this exercise is from Dummit and foote , page 117 , # 7.d
prove : a transitive group $G$ is primitive on $A$ iff for each $a \in A$ , the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$
...
2
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1answer
60 views
Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations
Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a
Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$
Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on
...
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
3answers
150 views
Exercises about group actions
I am having trouble with exercises from Chapter IV of Aluffi's Algebra: Chapter 0. After spending many hours on the following two, I guess I need some help:
Let $G$ be a finite group, and suppose ...
1
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1answer
71 views
determine stabilizer of an edge of the cube and its orbit
Let $G$ be the group of symmetries of the cube, and consider the action of $G$ on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of $G$.
...
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2answers
331 views
Orbit, stabilizer and fixed points of a group's action on left cosets by left multiplication
For this example - with hardly any steps - I'm not getting its conclusions with my work. Can someone please see where I might have gone wrong? I added the colors. Thank you.
Given Example
$G$ ...
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3answers
142 views
Faithfulness - Group Action on Left Cosets by Left Multiplication
A lecture stated that if $ H \leq \text{ group } G $ and $ G $ acts on $ \{gH\} = G/H $ by left multiplication, then this left multiplication action of $ G \text{ on } {gH} = G/H $ is faithful iff $ ...
5
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3answers
108 views
Group Action - Permutation on the Polynomial
I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...
2
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1answer
48 views
Virtually infinite cyclic groups act on a tree
A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers.
But the ...
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votes
1answer
86 views
Invariants of binary forms under a $\begin{pmatrix} 1& 1 \\ 0& 1 \end{pmatrix}$ action
The special linear group $\text{SL}_2(\mathbb{Z})$ of $2\times 2$ invertible matrices in $\mathbb{Z}$ acts on binary cubic forms $\{ax^3 + bx^2y + cxy^2 + dy^3\}$ by acting on the vector $(x,y)^T$. ...
2
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1answer
69 views
Motivation for the term “transitive” group action
I have two questions:
In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not ...
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0answers
87 views
Properly discontinuous action on a non-locally compact space
Let me begin with some definitions in order to avoid confusion.
An action of a group $G$ on a space $X$ is proper if the map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, gx)$ is proper, ...
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1answer
41 views
Definition of tautological action
What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions?
For reference the particular sentence is: "A ...
3
votes
2answers
100 views
Original papers on the subject of group actions
Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. ...
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votes
2answers
66 views
Composite group homomorphism between alternating groups
Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
3
votes
2answers
150 views
Transitive action of normal subgroup of the alternating group
everyone! Would anyone be willing to give me any sort of help with the following question?
Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the ...
4
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0answers
147 views
Semi-direct product isomorphic to direct product
I would like some help on the following problem from anyone who would like to help.
Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$.
The situation ...
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3answers
72 views
A step in the proof of Cauchy's theorem for groups
Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
0
votes
1answer
95 views
Injective Homomorphism on $S_n$
I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody ...
1
vote
1answer
62 views
Showing equality of two groups.
Consider $S_n$, the permutation group. Let $a\in S_n$. I want to show that if $a$ is an $n$ or an $n-1$ cycle, then $\langle a\rangle = \{c \in S_n : ca=ac\}$. Any help will be veru much appreciated.
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0answers
57 views
Isomorphism between G-equivariant bijections and Normalizer
If anyne could give me some help with this, it will be deeply appreciated:
Let $X$ be a set equipped with the action of some group $G$. Denote by $Aut_G(X)$ the set of $G-equivariant$ bijections $f: ...
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votes
1answer
61 views
A question on groups actions of permutation groups
This is the final step into completing a problem and I am a bit stuck. I need to show that:
Consider the action of $S_{n-1}$ on $S_n/S_{n-1}$ by left multiplication. Does this action have exactly two ...
1
vote
1answer
139 views
How to find isomorphism classes of transitive actions
I have not been able to find a proper definition of what an isomorphism class is (in the context of group theory). If one could define it properly for me and give me some help with the following two ...
1
vote
1answer
87 views
Two equal cyclic subgroups of $S_n$ have conjugate generators
The statement is relatively simple but the proof is giving me some trouble. Any help will be very much appreciated:
Let $a,b \in S_n$. Assuming $<a> = <b>$ show that $b$ is a conjugate of ...
0
votes
2answers
60 views
Normal subgroup and divisibility
I am very bad with problems involving divisibility of orders and such. If anyone can give me some help with the following problem, it will be very much appreciated:
Prove that every subroup $H$ of ...
0
votes
1answer
49 views
Rank is an invariant group action
I have to show following things and have no idea what I have to do at b)
a) $GL(n,K) \times K^{n \times m} \rightarrow K^{n \times m}: (g,A) \mapsto gA$ is a group action.
b) The Rank is an ...
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votes
1answer
52 views
Arc transitivity of the complete graph
Recall that a graph $G$ is arc transitive if the natural action of $\mathrm{Aut}(G)$ on $A(G) = \{ (u,v) | \{u,v\} \in E(G)\}$ is transitive.
In other words, given $(u,v),(u'.v') \in A(G)$ one finds ...
0
votes
0answers
79 views
Questions (doubts) on: Group Action on Manifolds
There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up:
Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...
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 ...
2
votes
2answers
195 views
Algebra - Infinite Dihedral Group
Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The ...
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0answers
44 views
G-H biset and left & right G-set
1) Prove that each left $G$-set $X$ can be turned into a right $G$-set by defining $x\sigma = \sigma^{-1}x$, and that every right $G$-set arises in this way.
I have the left action $X \times G \to X$ ...
2
votes
1answer
60 views
Confused over group action
Say I want to put action of $S_4$ on $V^{\otimes 4}$. If I want to act on the left, why can't I say
$$\sigma(v_1 \otimes v_2 \otimes v_3 \otimes v_4) = v_{\sigma(1)} \otimes \ldots \otimes ...
3
votes
1answer
133 views
Elliptic Points of Modular Group in Upper Half Plane
This is a very small question.
Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq ...
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 ...
2
votes
1answer
99 views
Extending a group action to a quotient group
If I have a cyclic group $G = (a)$ acting on an abelian group $A$, I need to define a natural action of $G$ on the quotient space $A/B$, where $B$ is a normal subgroup of $A$ with the property that ...
1
vote
0answers
36 views
A K-invariant submanifold of G-manifold and fundamental vector fields
Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$
$$
\underline{X}(p) := ...
3
votes
1answer
117 views
Circle Acting on Circle/Ball
Is much known about $S^1$-actions on the following simple spaces?:
1) $D^2$ the disk
2) More generally $D^n$ the n-ball
3) $S^1$ the circle
In particular, does every $S^1$-action on the disk (or ...
3
votes
2answers
223 views
Groups acting on topological spaces
In algebraic topology, we show that a group $G$ acting properly discontinously on a simply connected (and sufficently connected) topological space $Y$ is isomorphic to the fundamental group of $Y/G$.
...
4
votes
2answers
135 views
If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?
Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$.
Let $A$ be some ...
