Number of Sylow subgroups in finite groups

TL;DR

This paper investigates the divisibility properties of the number of Sylow p-subgroups in finite groups, reducing the problem to almost simple groups and providing a new proof for Navarro's theorem.

Contribution

It generalizes Navarro's result by reducing the divisibility question to almost simple groups and offers an alternative proof for the Navarro theorem.

Findings

Reduces the divisibility question to almost simple groups

Provides an alternative proof for Navarro's theorem

Enhances understanding of Sylow subgroup counts in finite groups

Abstract

Denote by $\nu_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $\nu_p(H)\leq\nu_p(G)$ for $H\leq G$, however $\nu_p(H)$ does not divide $\nu_p(G)$ in general. In this paper we reduce the question whether $\nu_p(H)$ divides $\nu_p(G)$ for every $H\leq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.