  This article explores the novel notion of gyrogroup actions, which is a
natural generalization of the usual notion of group actions. As a first step
toward the study of gyrogroup actions from the algebraic viewpoint, we prove
three well-known theorems in group theory for gyrogroups: the orbit-stabilizer
theorem, the orbit decomposition theorem, and the Burnside lemma (or the
Cauchy-Frobenius lemma). We then prove that under a certain condition, a
gyrogroup $G$ acts transitively on the set $G/H$ of left cosets of a
subgyrogroup $H$ in $G$ in a natural way. From this we prove the structure
theorem that every transitive action of a gyrogroup can be realized as a
gyrogroup action by left gyroaddition. We also exhibit concrete examples of
gyrogroup actions from the M\"obius and Einstein gyrogroups.
