A characterization of the 2-fusion system of L_4(q)

TL;DR

This paper characterizes the 2-fusion system of L_4(q) for certain q, advancing the classification of simple fusion systems at prime 2 by identifying unique structures based on specific conditions.

Contribution

It provides a new characterization of the 2-fusion system of L_4(q) using fusion analysis and transfer, contributing to the classification program of simple fusion systems.

Findings

F is uniquely determined as the 2-fusion system of L_4(q_1) for some q_1 ≡ 3 (mod 4)

The characterization relies on fusion analysis and transfer, avoiding heavy inductive methods

A related group-theoretic result is also established.

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 is 2-perfect with no nontrivial normal 2-subgroups, 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.