The Local Structure Theorem, the non-characteristic 2 case

Let p be a prime, G a finite K_p‐group, S a Sylow p‐subgroup of G and Q be a large subgroupof 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, O_p(L)≠1 and L≰N_G(Q).There is, however, one configuration where no structural informationabout L can be given using the methods in Meierfrankenfeld, Stellmacher and Stroth [Mem. Amer. Math. Soc. 242 (2016) 1147]. In this paperwe show that for p=2 this hypothetical configuration cannot occur. Weanticipate that our theorem will be used in the programme to revise theclassification of the finite simple groups.