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Component

This article is about a subgroup property related to the Classification of finite simple groups

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

Contents

History

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

Definition

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

Transitivity

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

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).

For full proof, refer: Components permute

Property operators

Right transiter

The right transiter of this property is: subnormal subgroup