# Layer

From Groupprops

(Redirected from Commuting product)

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Layer, all facts related to Layer) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

*This article is about a term related to the Classification of finite simple groups*

## Definition

### Symbol-free definition

The **layer** of a group (sometimes also called **commuting product**) is defined in the following equivalent ways:

- It is the commuting product of all components
- It is the unique largest semisimple normal subgroup

### Definition with symbols

The**layer**of a group , denoted is defined as:

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## Relation with other subgroup-defining functions

### Bigger subgroup-defining functions

- Generalized Fitting subgroup: The generalized Fitting subgroup of a group is the product of its Fitting subgroup and the layer. In symbols .