Wielandt's zipper lemma turns a failure of subnormality occurring only at the top level into uniqueness of a maximal overgroup. It is a key induction tool for proving that permutable subgroups are subnormal and that joins of subnormal subgroups remain subnormal. [@isaacs2008]
Definition
A unique maximal overgroup means one maximal subgroup \(M\) of \(G\) containing \(S\). It does not mean that \(S\) itself is maximal.
Core result
If two distinct maximal overgroups \(M,N\) existed, subnormal series inside them and the intersection theorem would push suitable normal closures of \(S\) into \(M\cap N\). Finiteness and maximality then force either \(\langle M,N\rangle<G\) or a subnormal series from \(S\) to \(G\), both contradictions. See Isaacs, Theorem 2.9 for the full double induction.
Structural properties
- The hypothesis controls every proper overgroup without requiring an explicit subnormal series to \(G\).
- A unique maximal overgroup is fixed under relevant conjugations and often becomes normal, producing the final contradiction.
- If \(S\) permutes with every conjugate \(S^g\), the zipper lemma helps prove that \(S\) is subnormal.
- In Wielandt's join theorem it removes the difficult case where two subnormal subgroups have no common proper overgroup.
- Infinite groups require an additional ascending-chain condition; finiteness supplies maximal overgroups and termination.
The lemma metaphorically closes local subnormal chains like a zipper. If the final step cannot close, all proper overgroups collapse to one maximal obstruction.
Example and boundary
Knowledge network
The subnormal core article proves the join theorem, permutable subgroups are a main application, and normal closure describes the conjugate-generated objects in the induction.
Proof and sources
The exact statement and complete finite double induction are Isaacs, Theorem 2.9; this article records its logical role, uses, and finiteness boundary. [@isaacs2008]
