Finite groups which are almost groups of Lie type in characteristic p

Let p be a prime. In this paper we investigate finite 𝒦_{2,p}-groups G which have a subgroup H ≤ G such that K ≤ H = N_G(K) ≤ Aut(K) for K a simple group of Lie type in characteristic p, and |G:H| is coprime to p. If G is of local characteristic p, then G is called almost of Lie type in characteristic p. Here G is of local characteristic p means that for all non-trivial p-subgroups P of G, and Q the largest normal p-subgroup in N_G(P) we have the containment C_G(Q) ≤ Q. We determine details of the structure of groups which are almost of Lie type in characteristic p. In particular, in the case that the rank of K is at least 3 we prove that G = H. If H has rank 2 and K is not PSL₃(p) we determine all the examples where G ≠ H. We further investigate the situation above in which G is of parabolic characteristic p. This is a weaker assumption than local characteristic p. In this case, especially when p ε {2,3}, many more examples appear.

In the appendices we compile a catalogue of results about the simple groups with proofs. These results may be of independent interest.