A 5-local identification of the monster

Abstract

Let G be a locally K-proper group, S ∈ Syl 5 (G), and Z = Z(S). We demonstrate that if N G (Z) ∼ 5 1+6 + .4 · J 2 .2 is 5-constrained and Z is not weakly closed in O 5 (N G (Z)) then G is isomorphic to the monster sporadic simple group.

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

“…However, it is very instructive to produce the extra handful of arguments and do without this sledge hammer. We mention here that there are odd p-local characterizations of some of the sporadic simple groups by Parker and Rowley [15] and Parker and Wiedorn [16] which do use the powerful K-proper assumption.…”

A 7-Local Identification of the Monster

“…Four examples occur with p = 5 and are listed as A 20 , A 21 , A 46 and A 53 in [22, Table 1.8]. These four amalgams are related to HN , BM , Ly, and M , respectively, and are the subject of [21], [23], and [24]. A further remarkable example which occurs in the monster has N β ∼ 13 1+2 + .12.Sym (4) and N α ∼ 13 2 .4.L 2 (13).2.…”

“…With respect to this amalgam the monster is of parabolic characteristic 13; unfortunately, as yet we have no idea how to characterize M from these two subgroups. The strategies used in [21], the present paper, [23] and [24] will not work in this particular case since the largest elementary abelian 13-subgroup in the centralizer of an involution in N α and N β has order 13. Thus the critical method of proof used in Propositions 5.6 and 6.5 to control O p (C G (t)) for an involution t fails.…”