Component

History
The notion of components is due to the work of Bender, (Gorenstein and Walter), and Wielandt.

Symbol-free definition
A subgroup of a group is termed a component if it is a quasisimple subnormal subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a component if $$H$$ is a quasisimple group and is also a subnormal subgroup in $$G$$.

Property theory
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

Self-permutability
Any two components of a group commute. In fact, the product of all components of a group is sometimes termed the commuting product (this is a subgroup-defining function).