Number of Sylow subgroups in finite groups

Abstract

Denote by ${\nu }_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that ${\nu }_p(H)\le {\nu }_p(G)$ for $H\le G$, 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 $H\le G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.