Finite groups of local characteristic p An Overview

A 3-local characterization of U6(2) and Fi22

“…Let E be an elementary abelian 2-group of order 2 18 . Then Aut(E) ∼ = GL 18 (2) contains a subgroup X similar to a 3-normalizer in U 6 (2).…”

“…Then Aut(E) ∼ = GL 18 (2) contains a subgroup X similar to a 3-normalizer in U 6 (2). Hence the semidirect product EX satisfies the hypothesis of Theorem 1 and, in this case, Z is weakly closed in S. In fact, if we were prepared to invoke the classification of finite simple groups, then we could apply [7,Remark 7.8.3] to see that if Z were weakly closed in S,…”

“…One application of Theorems 1 and 2 is to the end game of the investigation of groups of local characteristic p, which is being led by Meierfrankenfeld, Stellmacher and Stroth [18]. At the close of their work they will provide a collection of p-local subgroups forming possible amalgams in groups of local characteristic p. For the most part, they will then go on to recognize the possible groups via geometric methods.…”

A 7-Local Identification of the Monster

“…It is expected that the programme to identify the K-proper groups of local characteristic p orchestrated by Meierfrankenfeld, Stellmacher and Stroth (see [20]) will soon have a list of possible amalgams within such groups. Some of these amalgams will uniquely determine the target group via its p-local geometry (for example via the building if G is expected to be a Lie type group in characteristic p of rank at least 3).…”

On commutator subspaces of GF(pa)SL2(pa)-modules

“…It appears frequently as a subquotient in minimal parabolic subgroups of groups of Lie type in characteristic p and, on a wider canvas, often arises in classification problems. One reason why SL 2 (p a ) is encountered in such problems is that its natural 2-dimensional GF(p a )-module is a failure of factorization module-see for example [1], [3], [4]. Other modules for SL 2 (p a ) are also important by virtue of their appearance as chief factors in the unipotent radical of rank 1 parabolic subgroups.…”