Fusion Systems in Group Theory

Fusion systems provide an abstract framework for encoding the conjugacy relations of subgroups within a finite group, isolating the “p-local” behaviour at a fixed prime p. Rather than studying an entire group, one considers a category whose objects are the subgroups of a chosen Sylow p-subgroup and whose morphisms represent those group homomorphisms arising from conjugation in some ambient group. Central to this theory is the notion of a saturated fusion system, which satisfies axioms that mimic the Sylow and extension properties of finite groups. This categorical approach has led to the discovery of exotic fusion systems—structures that cannot be realised as the fusion of any finite group—and has deep connections with homotopy theory via classifying spaces. Classification efforts have identified families of fusion systems over small p-groups, and structural theorems establish criteria for reducibility, solvability and the existence of centric linking systems. The study of fusion systems has proved instrumental in understanding local-global phenomena in finite group theory, refining the classification of finite simple groups and offering new bridges between algebra, topology and representation theory. Concrete applications range from the analysis of group cohomology rings to conjectures on weights in block theory and to the construction of p-compact groups that mirror properties of compact Lie groups in a purely algebraic setting.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Fusion Systems in Group Theory publication trend

The graph below shows the total number of articles in fusion systems in group theory across all publications each year (not limited to Nature Index journals).

Technical terms

Fusion system: A category modelling conjugacy relations among subgroups of a given p-group, with morphisms induced by conjugation in an ambient group or abstractly defined.

Saturated fusion system: A fusion system satisfying axioms analogous to Sylow theorems, ensuring enough morphisms to extend automorphisms and to realise p-local structure.

Sylow p-subgroup: A maximal p-subgroup of a finite group, serving as the base object for defining a fusion system.

Exotic fusion system: A saturated fusion system that is not realised by conjugation in any finite group, highlighting genuinely new algebraic phenomena.

Strongly F-closed subgroup: A subgroup S₀ of the Sylow p-subgroup S such that any morphism in the fusion system sends S₀ into itself, used in solvability criteria.

References

1. Fusion systems on small p p -groups. Transactions of the American Mathematical Society (2012).
2. Extensions of linking systems and fusion systems. Transactions of the American Mathematical Society (2010).
3. The classification of 2 2 -compact groups. Journal of the American Mathematical Society (2008).
4. Criteria for supersolvability of saturated fusion systems. Journal of Algebra (2024).
5. Fusion systems on a Sylow p-subgroup of G2(p n ) or PSU4(p n ). Journal of Algebra (2023).
6. Weights for ℓ-local compact groups. Journal of Algebra (2023).