Finite Groups Which Are Almost Groups of Lie Type in Characteristic $\mathbf {p}$

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 = NG(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 nontrivial p-subgroups P of G, and Q the largest normal p-subgroup in NG(P) we have the containment CG(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 ? {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.