A characterization of the 2-fusion system of L4(q)

Abstract

We study a saturated fusion system F on a finite 2-group S having a Baumann component based on a dihedral 2-group. Assuming F = O 2 (F ), O 2 (F ) = 1, and the centralizer of the component is a cyclic 2-group, it is shown that F is uniquely determined as the 2-fusion system of L 4 (q 1 ) for some q 1 ≡ 3 (mod 4). This should be viewed as a contribution to a program recently outlined by M. Aschbacher for the classification of simple fusion systems at the prime 2. The corresponding problem in the component-type portion of the classification of finite simple groups (the L 2 (q), A 7 standard form problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy artillery. Thanks primarily to requiring the component to be Baumann, our main arguments by contrast require only 2-fusion analysis and transfer. We deduce a companion result in the category of groups.

Fusion systems with some sporadic J-components

“…and one sees that Q = C S (K) by combining Lemma 1.12(c) of [Lyn15] with Lemma 3.7(c) below. However, O 2 (C) is normal and self-centralizing in C C (K) by properties of the generalized Fitting subgroup, so that C C (K)/O 2 (C) is a group of outer automorphisms of the cyclic 2-group O 2 (C), and so is itself a 2-group.…”

“…In this paper, we classify saturated 2-fusion systems having a J-component isomorphic to the 2-fusion system of M 23 , J 3 , McL, or Ly under the assumption that the centralizer of the component is a cyclic 2-group. A similar problem for the fusion system of L 2 (q), q ≡ ±1 (mod 8) was treated in [Lyn15] under stronger hypotheses.…”

The Thompson–Lyons transfer lemma for fusion systems

“…To the extent that history is a guide, Theorem TL should serve as a key ingredient in the analysis of simple fusion systems at the prime 2, especially in the component-type portion of a program for the classification of simple 2-fusion systems outlined recently by Aschbacher et al; see [2,. Indeed, we postpone applications of Theorem TL to the companion [11], where we classify certain 2-fusion systems with an involution centralizer having a component based on a dihedral 2-group.…”

2-Fusion Systems with Components of Type SL2(q)

“…This is the fusion system version of a standard component of type SL 2 q , q odd. For another instance of work in support of this program, see the work of Lynd [11].…”