Number of Sylow subgroups in finite groups
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.