# Tagged Questions

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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### Is there a standard notion of a group “biaction”?

For a group $G$ there is a natural notion of left action on subsets of $G$ given by $g \triangleright H := gH$. But simultaneously, there is a natural right action as well: $H \triangleleft g := Hg$. ...
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### Is there a contractible space with a free circle action?

Question in title. Seems no to me (some vague intuition here about contracting orbits to a fixed point), but I can't prove it. I'd prefer to be wrong. (I'm curious because I am thinking about group ...
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### If a group acts properly and coboundedly on a hyperbolic space, each finite subgroup has a (uniformly) bounded orbit.

I am trying to solve the following problem: "Let $G$ be a group acting properly, coboundedly and by isometries on a hyperbolic space $X$. Show that there is a constant $C$ such that any finite ...
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### Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
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### Group action on algebra over a field defined on generators

Suppose $G$ is a group and $A$ is a finitely generated algebra over a field $\mathbb{k}$. Let $X=\{x_1,...,x_n\}$ be a set of generators for $A$, and suppose $G$ acts on $X$. Is this enough to define ...
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### The action of automorphisms on the Riemann sphere

If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without ...
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### How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...
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### Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
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### Categorical Quotient and group actions

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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### Showing that $x$ is an element of group $G$ by left multiplication

$G$ is a group and $H \leq G$ with $|G:H|=3$. Show that $x$ is an element of $H$ if $x \in G$ with $|x|=7$. Hint: let $\langle x \rangle$ act on $G/H$ by left multiplication and look at the orbits. ...
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### Do finite groups act admissibly on separated schemes of finite type over k

Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$...
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### An erroneous application of the Counting Theorem to a regular hexagon?

I'm trying to count the unique orbits of a regular hexagon such that each vertex is either Black or White and each edge is either Red, Gree, or Blue. The group I've chosen to act on the hexagon is the ...
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### In transitive (non-trivial) group action, there must be at least one group element without fixed point

Let a finite group $G$ act transitively on a finite set $S$ with $|S| \geq 2$. The problem is to show that not every $g \in G$ can have a fixed point in this action. I proved this on my own, but I'm ...
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### Interpretation of the join of two stabilizer subgroups

Let $G$ be the group acting on two sets $X, Y$. Let $G_x$ and $G_y$ be stabilizer subgroups of some elements $x \in X, y \in Y$. It is easy to see that $G_x \cap G_y = G_{(x,y)}$, when we combine two ...
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### Is $Y/K$ homeomorphic to $Y'$ as defined below -

Let $G$ be a topological group acting on a topological space $X$ in such a way that there are only finitely many orbits. We will fix points $x_1,\cdots,x_n\in X$ and let $X=\bigcup_{i=1}^n G\cdot x_i$ ...
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### Categorical Quotients and Group Actions on Varities

So I am given that Let $G = Z/dZ$ where d ≥ 1. Let w be a generator for G and let G act on $A^ {n+1}$ via $w(x_{0}, . . . , x_{n})$ = $(wx_{0}, . . . , wx_{n})$. How can I Show that the ...
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### References for the representation theory of $SU(2, 1)$
I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
We can regard a group action on a set as a functor $$F: BG \to Set\;,$$ where $BG$ is the category with one object and a morphism for each element of $G$, and $Set$ the category of sets. Now, is ...