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.

learn more… | top users | synonyms

19
votes
0answers
242 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 ...
13
votes
0answers
180 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 ...
12
votes
3answers
895 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
3answers
316 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 ...
12
votes
1answer
144 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
197 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
2answers
110 views

Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
11
votes
1answer
432 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 ...
10
votes
3answers
439 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
216 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
269 views

Distributing groups of objects into boxes

How can I enumerate the number of ways of distributing distinct groups of identical objects (but various cardinality) into $k$ boxes such that at most one box is empty $(1)$ and no combination of ...
10
votes
0answers
85 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
649 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 ↦ ...
9
votes
1answer
413 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
162 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
183 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 ...
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
179 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
81 views

Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?

Here $G$-sets denote the category of sets which have a left $G$-action. So the question is whether a functor $F \colon \text{$G$-sets} \to \text{$H$-sets}$ implies that we have an isomorphism of ...
7
votes
1answer
756 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
2answers
93 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
7
votes
3answers
169 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
2answers
128 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: ...
7
votes
1answer
262 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
137 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
172 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
243 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
5answers
919 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
295 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
1answer
339 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
72 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
141 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
219 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 ...
6
votes
0answers
48 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
770 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
105 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
519 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
86 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
90 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
88 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
188 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
137 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
2answers
126 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
3answers
85 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
1answer
230 views

The sum of orbit size of some element over the image of group “polynomial”

$\DeclareMathOperator{\orb}{orb}$ Say I have a group "polynomial", $p$, on $S_n$, that is $p(x)=a_1 x^{\epsilon_1}...a_n x^{\epsilon_n}$ for all $x \in S_n$, fixed $a_i \in S_n$ and fixed $\epsilon_i ...
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
252 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 ...