Conjugacy Classes and Orbits Visualization

The orbit of element x under conjugation is:

Orbit(x) = {gxg⁻¹ : g ∈ G}

This forms the conjugacy class of x.

Conjugation is the operation x ↦ gxg⁻¹ for some group element g.

Orbit of an element x is the set of all elements you can reach by conjugating x with every group element.

Conjugacy Class is exactly this orbit - elements that are "similar" under the group's structure.

Key Properties:

Conjugacy classes partition the group (every element belongs to exactly one class)
Elements in the same class have the same order
The identity element forms its own conjugacy class
In abelian groups, every element forms its own conjugacy class

🔄 Group Conjugation: Orbits & Conjugacy Classes