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)\leqslant\nu_{p}(G)} for {H\leqslant 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\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.

On the solvability of finite groups and the number of Sylow 2-subgroups

“…In 1995, Zhang [2], proved that a finite group G is p-nilpotent if and only if p is prime to every sylow number of G. In 2003, G. Navarro [3] proved that if G is p-solvable, then n p (H) divides n p (G) for every H ≤ G. In 2016 [4], Li and Liu classified finite non-abelian simple group with only solvable Sylow numbers. We say that a group G satisfies DivSyl(p) if n p (H) divides n p (G) for every H ≤ G. In 2018, Guo and E. P. Vdovin [5] generalized the results of G. Navarro, and proved that G satisfies DivSyl(p) provided every non-abelian composition factor of G satisfies DivSyl(p). Recently, Wu [6] proved that finite simple group does not satisfy DivSyl(p).…”