A quasisimple group is a perfect central cover of a nonabelian simple group. It permits a nontrivial center while retaining the rigidity that almost every proper normal subgroup is central. [@isaacs2008]
Definition
Quasisimple differs from almost simple: the former is a central extension, while the latter lies between a simple group and its automorphism group.
Core result
The image of a proper normal subgroup in simple \(L/Z(L)\) is trivial or full. The full case contradicts properness using \(L=L'\), so the subgroup is central. Perfectness descends to nontrivial quotients.
Structural properties
- A quasisimple group is perfect and has no nontrivial abelian quotient.
- Its central quotient is its unique nonabelian simple chief factor.
- Schur covering groups are standard sources of quasisimple central extensions.
- Quasisimple subnormal subgroups are exactly components.
- Distinct components can share central elements and form a central product.
Quasisimplicity lifts simple-group rigidity to a setting with controlled central error, ideal for commutator and central-product arguments.
Example and boundary
Knowledge network
Components are subnormal quasisimple groups, the layer is their product, and the generalized Fitting subgroup adjoins the nilpotent layer.
Proof and sources
Central quotients, perfectness, and normal-subgroup rigidity are Lemmas 9.1-9.2. [@isaacs2008]
