Wreath products, Sylow’s theorem and Fermat’s little theorem

Abstract

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 $alpha^{p−1}$ (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].