Normal closure enlarges a local object into normal structure visible to the whole group. It is generated by all conjugates and is the smallest way to force normality without discarding the chosen subset. [@isaacs2008]
Definition
The superscript \(G\) means generation by all \(G\)-conjugates, not one conjugacy class. A single conjugate subgroup is written \(H^g\).
Core result
The subgroup generated by all conjugates is invariant under conjugation and contains \(X\). Every normal subgroup containing \(X\) contains each \(x^g\) and therefore the generated subgroup. Compatibility with quotients follows because homomorphisms commute with conjugation.
Structural properties
- If \(H\triangleleft G\), then \(H^G=H\), and conversely this equality characterizes normality.
- For \(H\triangleleft\triangleleft G\), the closure can be built one level at a time along a subnormal series, but it need not preserve the isomorphism type of \(H\).
- If \(S\) is nonabelian simple and subnormal, distinct conjugates commute, so \(S^G\) is a direct product of isomorphic simple groups and is minimal normal.
- The derived subgroup \(G'\) is the normal closure of all commutators and the smallest normal subgroup with abelian quotient.
- Bartels' theorem relates closure under strong conjugacy to ordinary normal closure in finite groups.
Normal closure is both a lattice closure operator and a common language for passing from generators, local subgroups, and simple components to normal structure.
Example and boundary
Knowledge network
Subnormality controls growth along a closure chain, the socle collects minimal normal closures, and component theory treats closures of quasisimple subnormal subgroups.
Proof and sources
Conjugate generation and closures of subnormal simple groups occur in Sections 2A and 9D; quotient and strong-conjugacy behavior is in Theorems 9.28-9.30. [@isaacs2008]
