Definition:Orbit (Group Theory)/Definition 2
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Definition
Let $G$ be a group acting on a set $X$.
Let $\RR$ be the relation on $X$ defined as:
- $\forall x, y \in X: x \mathrel \RR y \iff \exists g \in G: y = g * x$
where $*$ denotes the group action.
From Group Action Induces Equivalence Relation, $\RR$ is an equivalence relation.
The orbit of $x$, denoted $\Orb x$, is the equivalence class of $x$ under $\RR$.
Also see
- Group Action Induces Equivalence Relation, which demonstrates the equivalence of these definitions.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.6$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Definition $10.12$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions