Wreath Products, Sylow’s Theorem and Fermat’s Little Theorem

The assertion that the number of p-Sylow subgroups in a finite group is ≡ 1 mod p, begs the natural question whether one may obtain the power ap−1 (for any (a, p) = 1) as the number of p-Sylow subgroups in some group naturally. Indeed, it turns out to be so as we show below. The construction involves wreath products of groups. Using wreath products, a different generalization of Euler’s congruence (and, a fortiori, of Fermat’s little theorem) was obtained in [1].