Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩

Let p p be a prime. In this paper we investigate finite K { 2 , p } \mathcal K_{\{2,p\}} -groups G G which have a subgroup H ≤ G H \le G such that K ≤ H = N G ( K ) ≤ Aut ⁡ ( K ) K \le H = N_G(K) \le \operatorname {Aut}(K) for K K a simple group of Lie type in characteristic p p , and | G : H | |G:H| is coprime to p p . If G G is of local characteristic p p , then G G is called almost of Lie type in characteristic p p . Here G G is of local characteristic p p means that for all nontrivial p p -subgroups P P of G G , and Q Q the largest normal p p -subgroup in N G ( P ) N_G(P) we have the containment C G ( Q ) ≤ Q C_G(Q)\le Q . We determine details of the structure of groups which are almost of Lie type in characteristic p p . In particular, in the case that the rank of K K is at least 3 3 we prove that G =