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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5
votes
5answers
306 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 ...
3
votes
1answer
116 views

How to define own group action in GAP?

I am beginner in GAP. I have a group and a set. I wish to define an action the group on the set in my own way and wish to calculate its orbits and stabilizers. Is it possible? What is process?
2
votes
1answer
431 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| = ...
1
vote
2answers
127 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 ...
11
votes
1answer
263 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
2answers
159 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 ...
6
votes
1answer
187 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 ...
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 ...
4
votes
3answers
392 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 ...
11
votes
3answers
483 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 ...
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 ...
3
votes
2answers
550 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 ...
5
votes
0answers
100 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 ...
3
votes
2answers
265 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 ...
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 ...
2
votes
1answer
113 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 ...
2
votes
2answers
341 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 ...
1
vote
1answer
21 views

Cardinality of rational exponentiation orbit space

Let $X=(0,\infty)$ be the set of positive real numbers. Let $G=\mathbb{Q}\backslash\{0\}$ be the multiplicative group of rational numbers. $G$ acts freely on $X$ by exponentiation: $r\cdot x=x^r$ for ...
1
vote
1answer
45 views

Is the stabilizer of an element $\delta$ in the stabilizer of $\omega$ in G equal to the pointwise stabilizer of $\{ \delta, \omega \}$

i.e., is $(G_{\delta})_{\omega} = G_{( \{\delta, \omega\} )}$? I know that \begin{eqnarray*} (G_{\delta})_{\omega} &=& \{ \forall g \in G_{\delta} \,|\, \omega^g = \omega \} \\ &=& ...
1
vote
0answers
69 views

Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
0
votes
1answer
32 views

the group acts faithfully on the line

Let $G$ be a group. $G$ acts faithfully on the line $\mathbb{R}$ by orientation preserving homeomorphism, then does it imply $G$ is left ordered, i.e. there is an order $<$ on $G$, and if $a<b$, ...
0
votes
1answer
36 views

Notation for permutation corresponding to the action of a group element

Let $G \times X \to X,\ \ (g,x) \mapsto g.x$ be an action of $G$ on $X$, i.e., $e.x = x$ for all $x \in X$; $gh.x = g.(h.x)$ for all $g \in G$, $x \in X$. For a fixed $g \in G$, how should I refer ...
0
votes
2answers
502 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 ...