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

Chris Parker, Gerald Pientka, Andreas Seidel, Gernot Stroth

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Abstract

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 non-trivial 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 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.
Original languageEnglish
JournalMemoirs of the American Mathematical Society
Publication statusAccepted/In press - 13 Apr 2021

Bibliographical note

Not yet published as of 10/05/2022.

Keywords

  • finite groups
  • embedding
  • identification of finite simple groups

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