Large subgroups of fusion systems and localities

Abstract: Saturated fusion systems are categories modeling properties of conjugacy of p-subgroups in finite groups. It was shown by Chermak that they correspond nicely to group-like structures called localities. In this paper we start to explore how concepts and results from a program of Meierfrankenfeld, Stellmacher and Stroth, aiming to reprove and generalize parts of the classification of the finite simple groups, translate to fusion systems and localities. Central in the program is the notion of a large $p$-subgroup. The presence of a large $p$-subgroup in a finite group turns out to be strong enough information to nearly classify the entire $p$-local structure, while also accommodating a very large class of groups of interest including many groups of Lie type in defining characteristic $p$. Utilizing the group-theoretical definition of a large $p$-subgroup as a blueprint, we define large subgroups of fusion systems and localities. We then analyze how the three definitions relate to each other, showing in particular that the newly defined notions behave well under the correspondence between saturated fusion systems and localities with certain properties. We further proceed with an example of how classification results from the program of Meierfrankenfeld et.al. translate to fusion systems and localities. In more detail, we give a new characterization of the $2$-fusion system of $\operatorname{Aut}(\operatorname{G}_2(3))$ following the strategy in a paper of Meierfrankenfeld and Stroth, where the group $\operatorname{Aut}(\operatorname{G}_2(3))$ is characterized in a similar fashion.

• The paper introduces a translation of large subgroup concepts from finite groups to fusion systems and localities.
• It establishes correspondence between large subgroups in groups, fusion systems, and localities, underscoring structural consistency.
• It applies the new framework to characterize the 2-fusion system of Aut(G2(3)), offering a novel approach for classification.

Large Subgroups in Fusion Systems and Localities

Introduction and Context

The paper "Large subgroups of fusion systems and localities" (2603.29607) develops a systematic foundation for the translation and analysis of "large subgroups"—a pivotal concept from finite group theory—within the context of saturated fusion systems and localities. The motivation is rooted in programs (by Meierfrankenfeld, Stellmacher, Stroth, and Aschbacher) aiming to rederive classification results of finite simple groups by recasting local group-theoretic arguments in the categorical setting of fusion systems, particularly for the prime $2$. The authors focus on three axes:

• Definitional transfer: Proposing natural analogues of large $p$-subgroups for fusion systems and localities, and analyzing their formal and structural properties.
• Correspondence: Proving that these notions behave well under the Chermak equivalence between saturated fusion systems and localities, ensuring the transfer of information between categorical and group-like settings.
• Application to classification: Demonstrating the practical power of these definitions by characterizing the $2$-fusion system of $Aut(G_2(3))$ in fusion-theoretic terms.

This contribution integrates modern finite local group theory with the abstract machinery of partial groups and localities, further bridging the gap between group and fusion system classification strategies.

Large Subgroups: Definitions and Fundamental Properties

In Finite Groups

A $p$-subgroup $Q$ of a finite group $G$ is large if it satisfies:

• Centralizer condition: $C_G(Q) \le Q$,
• $Q$-uniqueness property: For all $1 \ne U \le Z(Q)$, $p$0.

The presence of a large $p$1-subgroup controls the $p$2-local structure, paralleling the characteristic $p$3 configuration (all $p$4-local subgroups of characteristic $p$5).

In Fusion Systems and Localities

Analogous definitions are provided for fusion systems $p$6 on $p$7, and localities $p$8. A subgroup $p$9 is large in $2$0 if:

• $2$1,
• For all $2$2, $2$3.

For a locality, similar containment requirements hold, but at the level of partial or global normalizers in $2$4.

Structural Consequences

The paper establishes analogues of known group-theoretic properties for large subgroups in fusion systems and localities:

• The $2$5-subgroup of the normalizer of a large $2$6 is large.
• Large subgroups are always weakly closed (in their relevant context).
• Unions and products of large subgroups remain large under normalizing relations.
• In saturated fusion systems, further consequences include that every $2$7-local subsystem containing a large $2$8 is constrained, i.e., the fusion-theoretic version of parabolic characteristic $2$9.

These properties are derived, and their interplay with localization (e.g., linking localities, subcentric localities) and partial group theory is rigorously charted.

Compatibility and Correspondence: Groups, Fusion Systems, Localities

A principal task is verifying that the notion of "large" passes cleanly between groups, their fusion systems, and associated localities. For large subgroups $Aut(G_2(3))$0 in a group $Aut(G_2(3))$1, it is shown that $Aut(G_2(3))$2 remains large in $Aut(G_2(3))$3. The converse is subtler: one can construct examples (even with $Aut(G_2(3))$4) where $Aut(G_2(3))$5 is large in the fusion system but not in $Aut(G_2(3))$6, sometimes failing the $Aut(G_2(3))$7-uniqueness property or the self-centralizing property. This diagnosis is refined with precise group-theoretic equivalences (e.g., in terms of the strong closure of characteristic subgroups in normalizers).

The translation is codified in the context of localities. Here, ensuring the object set $Aut(G_2(3))$8 is sufficiently rich—e.g., being "Q-replete" (every nontrivial subgroup normalized by $Aut(G_2(3))$9 is in $p$0)—enables a full correspondence of "largeness." Subcentric linking localities play a critical role in this matching.

Application: Characterization of the $p$1-Fusion System of $p$2

A significant anchor for the theory is the translation of a classification theorem: a characterization of those saturated $p$3-fusion systems isomorphic to the one associated to $p$4. Abstracting an argument from Meierfrankenfeld and Stroth's group-theoretic paper, and after suitable technical refinements, the authors prove:

• Theorem: Let $p$5 be a saturated fusion system on $p$6 and $p$7 large in $p$8 with $p$9. Under the existence of a containing group $Q$0 of characteristic $Q$1 with prescribed normal $Q$2-subgroup and natural $Q$3-module structure, there is a fusion-theoretic isomorphism $Q$4.

The proof constructs and analyzes the relevant amalgam—subsystems, normalizers, and module action—entirely in the fusion/locality framework, sidestepping recourse to group-level Aschbacher-type amalgam criteria. The precise transfer of also possible gaps (noted in previous literature) is treated with a layer of abstraction, showing the robustness of the locality approach to such character-theoretic and amalgam-based characterizations.

Implications and Outlook

Theoretical Implications

• The definitions provide solid foundations for transposing large parts of local group-theoretic structure theory into the world of fusion systems and localities, with immediate applicability to classification programs (for both finite simple groups and saturated fusion systems).
• The compatibility across groups, saturated fusion systems, and localities, with precise control via object set choices, supports the development of locality-based analogues to $Q$5-local analysis, possibly even independent proofs of classification theorems.

Practical Implications

• This framework supports the analysis and potential identification of exotic fusion systems: large subgroups serve as structural constraints filtering the search space for such systems.
• The explicit characterization of $Q$6-fusion systems for almost simple groups (and possibly their automorphism extensions), via large subgroup structure, may drive future algorithmic and computational approaches to the recognition of fusion systems arising in finite group theory and algebraic topology.

Future Directions

• Developing explicit criteria and recognition theorems for saturated fusion systems at odd primes, beyond component type and current limitations. This will likely require a fusion-theoretic analog of yet broader parts of the finite group classification.
• Systematic analysis of exoticity for simple and non-simple cases, perhaps yielding filtrations or reduction algorithms for fusion system classification.
• Potential extensions to representation-theoretically motivated fusion systems (e.g., those arising from block theory), and their locality models.

Conclusion

The paper carefully constructs and systematically develops the concept of large subgroups in fusion systems and localities, transferring essential structural properties and demonstrating their power in classification tasks. The results codify the translation between group and fusion-theoretic frameworks, establishing key correspondences and verifying that large subgroups act as robust anchors for local analysis. The successful characterization of the $Q$7-fusion system of $Q$8 within this setting illustrates the utility and depth of the approach, setting a precedent for further theoretical development and cross-context recognition theorems in local theory, fusion systems, and finite simple group analysis.