What is a left group action?

A function $G\times X\to X$ satisfying $e\cdot x=x$ and $g\cdot(h\cdot x)=(gh)\cdot x$.

Why is the transformation associated with $g$ in a group action bijective?

Its inverse is the transformation associated with $g^{-1}$.

What is a free group action?

An action in which no nonidentity group element fixes any point.

What does the orbit–stabilizer theorem state for a finite group?

For $x\in X$, $|G\cdot x|=[G:G_x]=|G|/|G_x|$.

What is an equivariant map between two $G$-sets?

A map $f:X\to Y$ satisfying $f(g\cdot x)=g\cdot f(x)$ for all $g$ and $x$.

Abstract algebraLesson 1 of 1 in the course Group homomorphisms and quotients

Group Actions: Axioms, Orbits, Counting, and Representations

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A group action describes how the elements of a group transform the elements of a set while respecting the group’s algebraic structure. It provides a common language for studying symmetries, permutations, and transformations.

$e\cdot x=x,\qquad g\cdot(h\cdot x)=(gh)\cdot x.$

The first axiom says the identity group element leaves every point unchanged. The second says that applying $h$ and then $g$ has the same effect as applying the product $gh$ . A set equipped with a left action of $G$ is called a $G$ -set.

All flashcards

What is a left group action?: A function $G\times X\to X$ satisfying $e\cdot x=x$ and $g\cdot(h\cdot x)=(gh)\cdot x$ .
Why is the transformation associated with $g$ in a group action bijective?: Its inverse is the transformation associated with $g^{-1}$ .
What is a free group action?: An action in which no nonidentity group element fixes any point.
What does the orbit–stabilizer theorem state for a finite group?: For $x\in X$ , $|G\cdot x|=[G:G_x]=|G|/|G_x|$ .
What is an equivariant map between two $G$ -sets?: A map $f:X\to Y$ satisfying $f(g\cdot x)=g\cdot f(x)$ for all $g$ and $x$ .

Quiz questions

What condition defines a faithful group action?Answer: Only the identity element fixes every point of the set.
What distinguishes an $n$ -transitive action from an $n$ -homogeneous action?Answer: Transitivity concerns ordered lists; homogeneity concerns subsets.
A finite group of order $24$ acts on a set, and the orbit of $x$ has $6$ points. What is $|G_x|$ ?Answer: $4$
For a finite group $G$ acting on a finite set $X$ , how does Burnside’s lemma determine the number of orbits?Answer: It averages the fixed-point counts over all elements of $G$ .
Which equation expresses equivariance of a map $f:X\to Y$ between $G$ -sets?Answer: $f(g\cdot x)=g\cdot f(x)$

Based on the Wikipedia article Group action, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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