Definition:Orbit (Group Theory)/Set of Orbits
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Definition
Let $G$ be a group.
Let $X$ be a set
Let $*: G \times S \to S$ be a group action.
Let $\Orb x$ denote the orbit of $x \in X$.
From Group Action Induces Equivalence Relation, the relation $\RR_G$ defined as:
- $x \mathrel {\RR_G} y \iff y \in \Orb x$
is an equivalence relation.
The quotient set $X / \RR_G$ is called the set of orbits of $X$ under the action of $G$.
Also see
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54$