# Questions tagged [group-actions]

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 $2n$ act 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.

3,168 questions
Filter by
Sorted by
Tagged with
10 views

### Small examples of non-transitive Automorphism groups of Steiner Systems

I'm currently doing research for a bachelor's seminar talk. I have found a result from E. Mendelsohn, "On the groups of automorphisms of Steiner triple and quadruple systems" stating that ...
27 views

### An action of $\Gamma_0(N)$ having finitely many orbits

For positive integers $m$ and $N$ let \begin{align} M_{2, m, N}(\mathbf{Z}) = \bigg\{\gamma \in M_2(\mathbf{Z}) \; \bigg\vert \; \det(\gamma) = m, \gamma \equiv \bigg(\begin{matrix} \ast & \ast \\ ...
28 views

• 4,794
62 views

• 158
135 views

• 1,268
1 vote
32 views

### Induced action on fundamental group

I encountered this problem on a practice exam (without sample solution): Identify $S^2$ with $\mathbb{C} \cup \infty$. Let $X_n = S^2\backslash\mathbb{Z}/n\mathbb{Z}$ denote $S^2$ minus the N-th roots ...
• 383
62 views

### Alternative definitions outer semidirect product [closed]

I am working with outer semidirect products and have come across two definitions that are slightly different. Let $N,H$ be two groups and $\theta: H \to \text{Aut}(N)$ a group morphism. Then in both ...
• 415
1 vote
14 views

### Do amenable groups have invariant monotiles?

Let $G$ be a countable, discrete, amenable group, and let $B$ a finite subset. We say that $B$ is a monotile if $G$ is a disjoint union of translated copies of $B$. In "Monotilable Amenable ...
67 views

### What is the name of the group of all circular slice rotations on the sphere?

Consider the following group that naturally acts on the $2$-sphere. An individual generator works by selecting a circle $c$ on the surface of the $2$-sphere, and then rotating $c$ in the plane ...
• 17.5k
1 vote
45 views

99 views

### Prove $|M/G|=\sum_{x \in M} {\frac{1}{|G(x)|}}$

Let $G$ be a finite group operating on a finite set $M$. $|M/G|$ is the number of orbits. I want to prove that $|M/G|=\sum_{x \in M} {\frac{1}{|G(x)|}}$ This question is in preparation of Burnside-...
• 189