Question 865859
For example, let *[tex \large G = \{ x | x^4 = 1 \}](that is, the elements of G are 1, x, x^2, x^3 with x^4 = 1), and suppose G acts on *[tex \large S = \{1, x, x^2, x^3\}]. Then, for *[tex \large k \in S],


*[tex \large Orbit (k) = \{ gk | g \in G \}]
*[tex \large Stab (k) = \{ g \in G: gk = k \}]


It follows that for all *[tex \large k \in S], *[tex \large Orbit(k) = \{1, x, x^2, x^3 \}] and *[tex \large Stab(k) = \{1\}].


This satisfies the orbit-stabilizer theorem *[tex \large Orbit(k) \cdot Stab(k) = |G|] for *[tex \large k \in G].