The focal subgroup theorem was introduced in (Higman 1953) and is the "first major application of the transfer" according to (Gorenstein, Lyons & Solomon 1996, p. 90).
One also gets the following result: The focal subgroup of a finite group G with Sylow p-subgroup P is given by: where v is the transfer homomorphism from G to P/[P,P], (Isaacs 2008, Theorem 5.21, p. 165).
Interest in the hyperfocal subgroups was renewed by work of (Puig 2000) in understanding the modular representation theory of certain well behaved blocks.
The hyperfocal subgroup of P in G can defined as P∩γ∞(G) that is, as a Sylow p-subgroup of the nilpotent residual of G. If P is a Sylow p-subgroup of the finite group G, then one gets the standard focal subgroup theorem: and the local characterization: This compares to the local characterization of the focal subgroup as: Puig is interested in the generalization of this situation to fusion systems, a categorical model of the fusion pattern of a Sylow p-subgroup with respect to a finite group that also models the fusion pattern of a defect group of a p-block in modular representation theory.
In fact fusion systems have found a number of surprising applications and inspirations in the area of algebraic topology known as equivariant homotopy theory.
For instance, the influential work (Alperin 1967) develops the idea of a local control of fusion, and as an example application shows that: The choice of the family C can be made in many ways (C is what is called a "weak conjugation family" in (Alperin 1967)), and several examples are given: one can take C to be all non-identity subgroups of P, or the smaller choice of just the intersections Q = P ∩ Pg for g in G in which NP(Q) and NPg(Q) are both Sylow p-subgroups of NG(Q).
138–165), all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of splitting called p-nilpotence.
In other words, finite groups with quasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns.