A sieve formula for chains of $p$-subgroups

TL;DR

This paper introduces a sieve formula for chains of p-subgroups in finite groups, generalizing Sylow-Frobenius theorem and inspired by set theory sieve concepts.

Contribution

It establishes a new congruence involving chains of p-subgroups, extending classical results with a novel approach based on Wielandt's methods.

Findings

Derived a congruence involving p-subgroup chains

Generalized Sylow-Frobenius theorem

Connected sieve formula to set theory concepts

Abstract

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.