On Some Generalizations of Fermat's, Lucas's and Wilson's Theorems

“Never underestimate a theorem that counts something!” – or so says J. Fraleigh in his classic text [2]. Indeed, in [1] and [4], the authors derive Fermat’s (little), Lucas’s and Wilson’s theorems, among other results, all from a single combinatorial lemma. This lemma can be derived by applying Burnside’s theorem to an action by a cyclic group of prime order. In this note, we generalize this lemma by applying Burnside’s theorem to the corresponding action by an arbitrary finite cyclic group. We revisit the constructions in [1] and [4] and derive three divisibility theorems for which the aforementioned classical theorems are, respectively, the cases of a prime divisor. Throughout, n and p denote positive integers with p prime and Zn denotes the cyclic group of integers under addition modulo n.