Permutability uses equality of set products \(HK=KH\), a condition weaker than mutual normalization. It does not visibly provide a normal series, but Wielandt's zipper lemma turns conjugate-permutability into subnormality in finite groups. [@isaacs2008]
Definition
When \(HK=KH\), the set product is a subgroup. This is not equivalent to \(H\leq N_G(K)\) or \(K\leq N_G(H)\).
Core result
Induction on \(|G|\) makes \(S\) subnormal in every proper overgroup. If it were not subnormal in \(G\), the zipper lemma would give a unique maximal overgroup. Conjugate-permutability makes that overgroup invariant under enough conjugations to force it to be all of \(G\), a contradiction. See Theorems 2.8-2.10.
Structural properties
- Every normal subgroup permutes with every subgroup, but the converse is false.
- Permuting finite subgroups satisfy \(|HK|=|H||K|/|H\cap K|\).
- If their orders are coprime, a permuting product has semidirect-product form, although both factors need not be normal.
- Conjugate-permutability forces subnormality but generally not normality.
- Homomorphic images of permutable subgroups remain permutable, with corresponding inverse-image statements.
Permutability lies between normality and arbitrary subgroup position and is useful for converting product factorizations into subnormal structure.
Example and boundary
Knowledge network
The subnormal core article proves the conclusion, the zipper lemma is the induction engine, and normalizers express the stronger setwise-preservation condition.
Proof and sources
The full proof that conjugate-permutability implies subnormality is Isaacs, Theorems 2.8-2.10; this article preserves its exact hypothesis and limits. [@isaacs2008]
