Fusion describes how conjugacy classes inside a subgroup merge in a larger group. Control of fusion says that an intermediate subgroup already realizes every such merger. [@isaacs2008]
Definition
Fusion may refer to elements, subgroups, or injective homomorphisms. This entry begins with element fusion in \(P\).
Core result
The focal subgroup is generated by conjugacy differences inside \(P\). Fusion control replaces every \(G\)-difference by an \(H\)-difference. Equality of focal subgroups then identifies the transfer kernels and maximal abelian \(p\)-quotients. See Corollary 5.22.
Structural properties
- \(G\) always controls its own fusion; control by \(P\) itself is a strong local condition.
- If \(P\) is an abelian Sylow subgroup, then \(N_G(P)\) controls fusion in \(P\).
- Control of fusion implies control of transfer, but the converse need not hold.
- Fusion can be generated from conjugation data in Sylow normalizers, centralizers, and other local subgroups.
- Saturated fusion systems abstract these conjugacy monomorphisms into an independent category.
Fusion control is a strong localization of global conjugacy. Transfer retains only its abelianized shadow.
Example and boundary
Knowledge network
The focal theorem converts fusion differences into a derived intersection, transfer control is the weaker consequence, and normalizers are the usual candidates.
Proof and sources
Definitions, the abelian-Sylow normalizer result, and the transfer consequence are in Sections 5C-5D. [@isaacs2008]
