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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19
votes
0answers
225 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 ...
12
votes
3answers
695 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 ...
12
votes
1answer
139 views

Is $GL_2(\mathbb Z)\cdot X$ a dense subset of $\mathbb R^2$?

We know that the set $D=\{a+b\sqrt{2} \mid a,b\in \mathbb Z\}$ is dense in $\mathbb R$ because $D$ is a subgroup of $(\mathbb R,+)$ that is not of the form $\alpha \mathbb Z$. So, the following set ...
11
votes
2answers
190 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 ...
11
votes
3answers
295 views

Elementary Combinatorial Proofs using group action

In trying to prove that the number of spanning trees in $K_5$ is $125$ I adopted the following method: Let $S$ be the set of all such spanning trees and let $S_5$ act in a natural way on $S$. Now ...
11
votes
1answer
394 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 ...
11
votes
0answers
113 views

Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
10
votes
3answers
435 views

Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
10
votes
1answer
214 views

Writing $G/A\times G/B$ explicitly as union of orbits

Let $G$ be a finite abelian group, and let $A$ and $B$ be subgroups. I'm interested in $G/A\times G/B$ with its natural $G$-set structure. In $G/A\times G/B$, the stabilizer of any element is $A\cap ...
10
votes
0answers
84 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 ...
9
votes
1answer
354 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 ...
9
votes
1answer
156 views

The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
8
votes
1answer
589 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 ↦ ...
8
votes
2answers
1k views

Kernel of Group Action

this is my first post here. I have a question regarding a proof in Algebra by Hungerford: Let $G$ be a group and $H$ a subgroup of $G$. Let $S$ be the set of all cosets of $H$, where $G$ acts on ...
8
votes
0answers
176 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
7
votes
1answer
652 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!
7
votes
3answers
166 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 ...
7
votes
1answer
257 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 ...
7
votes
1answer
156 views

Understanding what an action is?

This is a very simple question, and I am quite embarrassed to ask it! I'm trying to understand what an action is in general, and perhaps the best place to start is to try and outline my current ...
7
votes
1answer
131 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 $(x,y)^T$. ...
7
votes
1answer
65 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
7
votes
1answer
169 views

Why are they called orbits?

When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\} $$ that are called orbits. Although, the only reason I find convincing for that name is that in ...
6
votes
5answers
239 views

What does it mean for a group to “act algebraically”?

I'm reading the paper How to use finite fields for problems concerning infinite fields, by Serre. In Theorem 1.2 on page 1, he says Let $G$ be a finite $p$-group acting algebraically... In the ...
6
votes
4answers
824 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 ...
6
votes
4answers
277 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 ...
6
votes
2answers
112 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
6
votes
1answer
308 views

If a finite group $|G|$ acts transitively on a set $X$ with $|X|=2^n$, $n \geq 1$, then $G$ has an involution with no fixed points

Let $G$ be a finite group acting transitively on a set $X$, where $|X| = 2^n$ for some $n \geq 1$. Show that some element of $G$ acts as an involution with no fixed points. While it is fairly easy ...
6
votes
1answer
67 views

Number of ways to pick N numbers from 0,1,…,N-1, with possible duplication, with sum equal 0 mod N

We have the numbers $0,1,2,....,N-1$ in $\mathbb Z_N.$ I want to pick $N$ numbers from these. These are the rules: Duplication may occur We don't care about ordering, $00041$ is equivalent to ...
6
votes
1answer
136 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
94 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 ...
6
votes
0answers
45 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
3answers
716 views

Without using Sylow: Group of order 28 has a normal subgroup of order 7

Prove that a group of order 28 has a normal subgroup of order 7. How can I prove this without using Sylow's theorem? I know by Cauchy’s theorem, there exists an $x\in G$ with order 7, now I just ...
5
votes
4answers
104 views

On what sets can $\mathfrak{S}_n$ act transitively?

I would like to know $\mathfrak{S}_n$ could act faithfully transitively on sets with $m$ elements, with $m > n$. I know that it is not possible if $m = n+1$ except for $n = 5$. Any ideas ?
5
votes
5answers
479 views

Poincaré's theorem about groups

Let $G$ be a group and $H<G$ such that $[G:H]<\infty$. There exists a subgroup $N\triangleleft G$ such that $[G:N]<\infty$. I have to show this fact (that according to my book is due to ...
5
votes
1answer
85 views

Recovering a group action from sizes of orbits of individual elements

Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...
5
votes
3answers
88 views

Involution on Cantor space with exactly one fixed point

Let $X=\{0,1\}^{\mathbb{N}}$ be the Cantor space. What is an example of a continuous map $\sigma : X \to X$ with $\sigma^2=\mathrm{id}$ and $\# \{x \in X : \sigma(x)=x\} = 1$? This has to exist, ...
5
votes
2answers
86 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
5
votes
1answer
176 views

$Q_8$ is isomorphic to a subgroup of $S_8$ but not to asubgroup of $S_n$ for $n\leq 7$.

Question is to Prove that : $Q_8$ is isomorphic to a subgroup of $S_8$ but not isomorphic to a subgroup of $S_n$ for $n\leq 7$. I see that $Q_8$ is isomorphic to subgroup of $S_8$ by left ...
5
votes
2answers
136 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 ...
5
votes
3answers
65 views

How many faces, edges and vertices are fixed when $S_4$ permutes the diagonals of a cube?

Consider the action of $S_4$ on a cube, where it acts by permuting the long diagonals. The conjugacy classes of $S_4$ are denoted by $id$, (12), (123), (1234) and (12)(34). I want to know the number ...
5
votes
2answers
78 views

Difference between Stabilizer and Centralizer?

I know that the Centralizer of an element $a$ in a Group $G$ is defined as follows $$C_G(a) = \{ g \in G \space | \space ga = ag \}$$. It can also be defined as follows $$C_G(a) = \{ g \in G \space ...
5
votes
1answer
49 views

Orbit space of $S^n \times S^n$ under the antipodal action

Write $S^n$ for the $n$-dimensional sphere, the space of vectors of length $1$ in $(n+1)$-dimensional Euclidean space. Consider the antipodal action on $S^n$, i.e. the action of $\mathbb{Z}_2$ given ...
5
votes
1answer
224 views

Two subgroups $H_1, H_2$ of a group $G$ are conjugate iff $G/H_1$ and $G/H_2$ are isomorphic

Let $H_1$ and $H_2$ be subgroups of some group $G$. Prove that the left $G$-sets $G/H_1$ and $G/H_2$ are isomorphic (as left $G$-sets) iff the subgroups $H_1$ and $H_2$ are conjugate. If $H_1$ ...
5
votes
1answer
136 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 ...
5
votes
1answer
86 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 ...
5
votes
1answer
711 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 ...
4
votes
3answers
59 views

is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
4
votes
2answers
112 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 ...
4
votes
2answers
150 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 ...
4
votes
3answers
76 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 ...