# Simple group operator

Template:Subgroup-to-group property operator

## Contents

## Definition

### Symbol-free definition

The simple group operator is an operator from the collection of trim subgroup properties to the group property space. It acts as follows: given a trim subgroup property , a group is termed -simple if it is nontrivial and has no proper nontrivial subgroup satisfying .

(A subgroup property is trim if it is satisfied by both the whole group and the trivial subgroup).

Note that the trivial group is not considered -simple.

## Examples

- Normal subgroup gives rise to simple group
- Characteristic subgroup gives rise to characteristically simple group
- Direct factor gives rise to directly indecomposable group
- Retract gives rise to semidirectly indecomposable group
- Ascendant subgroup gives rise to strictly simple group
- Serial subgroup gives rise to absolutely simple group

## Facts

### Simple-complete property

A trim property is termed simple-complete if every group can be embedded as a proper subgroup of a -simple group.

### Core operator

Suppose the given property is trim and join-closed. Then, in a -simple group, the -core of any proper subgroup is trivial. That is, every proper subgroup is -core-free.

If is a simple-complete property, then every subgroup of a group is **potentially core-free** with respect to .

### Closure operator

Suppose is intersection-closed and trim. Then, in a -simple group, the -closure of any nontrivial subgroup is the whole group. Equivalently, every nontrivial group is contra-.

Thus, if is a simple-complete property, then every nontrivial subgroup of a group is '*potentially contra*.