This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): quasisimple group
View a complete list of such conjunctions
Definition with symbols
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
In fact, any component of a subnormal subgroup is a component of the whole group.
Intermediate subgroup condition
A component of a group need not be a component in every intermediate subgroup. However, it is true that a component of a group is a component in every intermediate subnormal subgroup.
Pointwise permutability with subnormal subgroups
Given a component and a subnormal subgroup, either of the following is true:
- The component lies inside the subnormal subgroup
- Every element in the component commutes with every element inside the subnormal subgroup
For full proof, refer: Component commutes with or is contained in subnormal subgroup
For full proof, refer: Components permute