A sieve formula for chains of $p$-subgroups

Given a finite group $G$ and a prime $p$, we establish the sieve formula, which is a congruence containing as summands numbers of chains of $p$-subgroups of $G$ of certain orders. This generalises the Theorem of Sylow-Frobenius, using Wielandt’s approach. Its name stems from the sieve formula from set theory because of formal similarities.