Characterization of the orbits of a group action as equivalence classes

Let H be a group acting on a set A. Prove that the relation on A defined by a \sim b if and only if a = h \cdot b for some h \in H is an equivalence relation. (For each x \in A, the equivalence class of x under \sim is called the orbit of x under H.)

We need to show that the relation \sim is reflexive, symmetric, and transitive.

  1. For all x \in A, we have x = 1 \cdot x so that x \sim x.
  2. Suppose x \sim y; then we have x = h \cdot y for some h \in H. Then y = h^{-1} \cdot x, so that y \sim x.
  3. Suppose x \sim y and y \sim z. Then there exist h,k \in H such that x = h \cdot y and y = k \cdot z. Then x = h \cdot y = h \cdot (k \cdot z) = (hk) \cdot z, so that x \sim z.

Thus \sim is an equivalence relation.

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