We prove that if G is a finite almost simple group, having socle of Lie type of rank r, then the number of maximal subgroups of G is at most C r- 2 / 3 | G |, where C is an absolute constant. This verifies a conjecture of Wall for groups of sufficiently large rank. Using this we prove that any finite group G has at most 2 C | G |3 / 2 maximal subgroups.
Bibliographical noteFunding Information:
✩ The second author was supported by grants OTKA T049841 and NK62321. The third author acknowledges the support of an EPSRC Visiting Fellowship. * Corresponding author. E-mail address: firstname.lastname@example.org (A. Shalev).
- Finite groups
- Finite simple groups
- Maximal subgroups