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

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  • Martin-Luther-Universität Halle-Wittenberg
  • Martin-Luther University Halle-Wittenberg


Let p be a prime. In this paper we investigate finite \mathcal K_{\{2,p\}}-groups G which have a subgroup H \le G such that K \le H = N_G(K) \le \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)\le 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 \PSL_3(p) we determine all the examples where G \ne 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 \in \{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
Number of pages203
JournalMemoirs of the American Mathematical Society
Publication statusAccepted/In press - 13 Apr 2021