Burnside's theorem turns a very local centralization condition into a global normal \(p\)-complement and is one of the classical applications of transfer. [@isaacs2008]
Definition
The condition is also written \(N_G(P)=C_G(P)\), since \(P\leq N_G(P)\) and the normalizer induces no nontrivial conjugation on \(P\).
Core result
Evaluate transfer \(v:G\to P/P'=P\) on \(P\) by coset orbits. The centralization condition controls all nonprincipal orbit contributions and makes \(v|_P\) a power map with exponent \(|G:P|\), hence an automorphism. Transfer is surjective, and its kernel is a normal \(p'\)-subgroup of index \(|P|\).
Structural properties
- The hypothesis makes \(P\) abelian, but abelianness alone does not place \(P\) in the center of its normalizer.
- The transfer kernel has index \(|P|\) and therefore order coprime to \(p\).
- The theorem is a strong local special case of broader Frobenius normal-complement criteria.
- If \(N_G(P)=P\) and \(P\) is abelian, the hypothesis holds.
- The conclusion gives \(G=K\rtimes P\) with the unique normal \(p\)-complement \(K\).
The condition removes normalizer fusion on the Sylow subgroup, making transfer on \(P\) essentially a coprime power automorphism.
Example and boundary
Knowledge network
The transfer core article supplies the calculation, normalizers express the local hypothesis, and Frobenius gives a broader local criterion.
Proof and sources
The transfer proof and kernel-order calculation are Isaacs, Theorems 5.13-5.14; this entry records the complete route. [@isaacs2008]
