The Local Structure Theorem, the non‐characteristic 2 case

Abstract

Let p be a prime, G a finite Kp‐group, S a Sylow p‐subgroup of G and Q be a large subgroup of G in S. The aim of the Local Structure Theorem [Mem. Amer. Math. Soc. 242 (2016) 1147] is to provide structural information about subgroups L with S⩽L, Opfalse(Lfalse)≠1 and L≰NGfalse(Qfalse). There is, however, one configuration where no structural information about L can be given using the methods in Meierfrankenfeld, Stellmacher and Stroth [Mem. Amer. Math. Soc. 242 (2016) 1147]. In this paper we show that for p=2 this hypothetical configuration cannot occur. We anticipate that our theorem will be used in the programme to revise the classification of the finite simple groups.