Given a group action of a group \(G\) and a set \(X\). For any given \(x\in X\), the set \(G\cdot x = \{g\cdot x\mid g\in G\}\subseteq X\) is called orbit of \(x\).
Examples Link to heading
- \(\GL(n)\) on \(\mathbb{R}^n\): \(\mathbb{R}^n\setminus \{0\}\), \(\{0\}\)
- \(\O(n)\) on \(\mathbb{R}^n\): \(\{0\}\), all spheres
- \(\mathbb{R}\setminus \{0\}\) on \(\mathbb{R}^n\setminus \{0\}\): lines through origin with origin removed