# 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 of a group is termed a **component** if is a quasisimple group and is also a subnormal subgroup in .

## 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 transitiveABOUT 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*