Number of Sylow subgroups in finite groups

Abstract

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