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Group Actions

algebragroup theoryUpdated

An action translates the abstract multiplication of a group into transformations of a set.

The orbit of xx is the set of points reachable from it:

Orb⁡(x)={g⋅x:g∈G}.\operatorname{Orb}(x)=\{g\cdot x:g\in G\}.

The stabilizer is the subgroup of transformations fixing xx:

Stab⁡(x)={g∈G:g⋅x=x}.\operatorname{Stab}(x)=\{g\in G:g\cdot x=x\}.

Their sizes are related by the Orbit–Stabilizer Theorem.