Group theory and generalizations Proposition

Orbits partition the set

Statement

Let a group $G$ act on a set $X$ . The relation $x \sim y$ if and only if $y = g \cdot x$ for some $g \in G$ is an equivalence relation on $X$ . Its equivalence classes are the orbits, so each point of $X$ lies in exactly one orbit.

Remarks

Reflexivity uses $e \cdot x = x$ , symmetry uses $x = g^{-1} \cdot (g \cdot x)$ , and transitivity uses $h \cdot (g \cdot x) = (hg) \cdot x$ . If $X$ is finite, then $|X|$ is the sum of the sizes of the distinct orbits. The cosets of a subgroup and the conjugacy classes of a group are examples of orbits.

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Group concept
Group homomorphism definition
Symmetric group definition
Group action definition
Orbit definition
Orbits partition the set statement

Tag	`0ZC7FYE0NZ`
URI	`m2://0ZC7FYE0NZ`
Kind	statement, proposition
Writing	W0 stub
MSC	20B05
Status	This copy of the site has no build data. Run `m2 status 0ZC7FYE0NZ`.
Uses	Orbit, Group action

Orbits partition the set

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