[Paper Review] Reduced fusion systems over 2-groups of small order

Kasper K. S. Andersen, Bob Oliver+1 more|arXiv (Cornell University)|Jun 16, 2016

Finite Group Theory Research24 references3 citations

TL;DR

This paper proves that all saturated fusion systems over 2-groups of order at most 2^9 are realizable, by showing that every reduced, indecomposable fusion system over such groups is the fusion system of a finite simple group and is tame. The proof relies on a computational classification of 2-groups satisfying criteria derived from Bender’s theorem on strongly 2-embedded subgroups, demonstrating that only Sylow 2-subgroups of finite simple groups or product groups support reduced fusion systems in this range.

ABSTRACT

We prove, when $S$ is a $2$-group of order at most $2^9$, that each reduced fusion system over $S$ is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a $2$-group of order at most $2^9$ is realizable. What is most interesting about this result is the method of proof: we show that among $2$-groups with order in this range, the ones which can be Sylow $2$-subgroups of finite simple groups are almost completely determined by criteria based on Bender's classification of groups with strongly $2$-embedded subgroups.

Motivation & Objective

• To determine whether all saturated fusion systems over 2-groups of order at most 2^9 are realizable.
• To classify reduced, indecomposable fusion systems over small 2-groups and show they arise from finite simple groups.
• To establish tameness of reduced fusion systems in this range, leveraging prior results linking tameness to realizability.
• To identify which 2-groups of order ≤2^9 can support reduced fusion systems, using structural constraints from Bender’s classification of groups with strongly 2-embedded subgroups.

Proposed method

• A computer-assisted search using Magma and GAP to identify all 2-groups of order ≤2^9 satisfying specific structural conditions necessary for supporting reduced fusion systems.
• Application of Alperin’s fusion theorem to decompose morphisms in fusion systems into compositions of automorphisms of S and restrictions from essential subgroups.
• Identification of 'critical' subgroups (potential essential subgroups) in each 2-group, based on criteria derived from Bender’s theorem on strongly 2-embedded subgroups.
• Computation of possible automorphism groups of critical subgroups to determine possible fusion systems over each 2-group.
• Use of group-theoretic and modular representation-theoretic techniques (e.g., analyzing action on F_2[G]-modules) to rule out exotic or non-tame systems.
• Verification that only Sylow 2-subgroups of finite simple groups or products of smaller groups satisfy the necessary conditions, with the latter ruled out as non-reduced or non-indecomposable.

Experimental results

Research questions

• RQ1 Which 2-groups of order at most 2^9 can support a reduced fusion system?
• RQ2 Are all reduced fusion systems over 2-groups of order ≤2^9 tame, and if so, does tameness imply realizability?
• RQ3 Can the classification of groups with strongly 2-embedded subgroups by Bender be used to restrict the list of candidate 2-groups that support exotic or non-realizable fusion systems?
• RQ4 To what extent do Sylow 2-subgroups of finite simple groups coincide with the 2-groups that support reduced fusion systems in this order range?
• RQ5 What structural properties distinguish 2-groups that support reduced fusion systems from those that do not?

Key findings

• Among 2328 groups of order 2^7, only 9 satisfy the necessary conditions to support a reduced fusion system, 6 of which are Sylow 2-subgroups of finite simple groups.
• Among 56,092 groups of order 2^8, only 20 satisfy the conditions, 6 of which are Sylow 2-subgroups of finite simple groups.
• Among approximately 10^7 groups of order 2^9, only 34 satisfy the conditions, 10 of which are Sylow 2-subgroups of finite simple groups.
• All reduced, indecomposable fusion systems over 2-groups of order ≤2^9 are isomorphic to the fusion system of a finite simple group.
• All such fusion systems are tame, and hence realizable, as a consequence of prior results linking tameness to realizability.
• The only 2-groups that support reduced fusion systems in this range are either Sylow 2-subgroups of finite simple groups or products of smaller groups, with the latter excluded from supporting indecomposable reduced systems.