Definition:Orbit (Group Theory)/Definition 1
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Definition
Let $G$ be a group acting on a set $X$.
The orbit of an element $x \in X$ is defined as:
- $\Orb x := \set {y \in X: \exists g \in G: y = g * x}$
where $*$ denotes the group action.
That is, $\Orb x = G * x$.
Thus the orbit of an element is all its possible destinations under the group action.
Also see
- Group Action Induces Equivalence Relation, which demonstrates the equivalence of these definitions.
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Definition $10.14$