Tate's theorem is a 'supercharger' for transfer: once an intermediate subgroup controls the maximal abelian \(p\)-quotient, it also controls the smallest normal subgroup of \(p\)-power index. [@isaacs2008]
Definition
For \(P\leq H\leq G\) with \(P\) Sylow, the theorem compares each kernel with its intersection with \(H\).
Core result
The proof analyzes Frattini quotients and focal subgroups, lifting control of an abelian \(p\)-quotient one central layer at a time through finite \(p\)-group quotients. See Isaacs, Theorem 5.28.
Structural properties
- If \(H\) controls transfer and has a normal \(p\)-complement, then \(G\) has a normal \(p\)-complement.
- The theorem strengthens a kernel-intersection equality and does not assert control of all fusion.
- It upgrades abelianized detection to arbitrary finite \(p\)-group quotients.
- The Frattini quotient is the key bridge from a nonabelian \(p\)-quotient to its first abelian layer.
- Fusion systems have Tate-type hyperfocal generalizations.
Every finite \(p\)-group has nontrivial center and Frattini quotient, allowing control of the first abelian layer to be lifted successively.
Example and boundary
Knowledge network
Transfer control is the input, a normal p-complement is a common output, and focal/Frattini layers supply the proof mechanism.
Proof and sources
The theorem and converse remark are Isaacs, Theorem 5.28; the long induction is cited rather than duplicated. [@isaacs2008]
