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9781470467296 Academic Inspection Copy

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

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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.
Chris Parker, University of Birmingham, United Kingdom. Gerald Pientka, Halle, Germany. Andreas Seidel, Magdeburg, Germany. Gernot Stroth, Universitat Halle-Wittenberg, Germany.
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