# Splitting-simple group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of simple group|Find other variations of simple group |

## Contents

## Definition

A nontrivial group is said to be **splitting-simple** or **semidirectly indecomposable** or **inseparable** if it satisfies the following equivalent conditions:

- It cannot be expressed as a semidirect product of nontrivial groups
- It has no proper nontrivial retract
- It has no proper nontrivial complemented normal subgroup

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

simple group | nontrivial and no proper nontrivial subgroup is normal | |FULL LIST, MORE INFO | ||

quasisimple group | perfect, and inner automorphism group is simple | |FULL LIST, MORE INFO |

### Stronger properties conditional to nontriviality

The following properties are stronger, assuming the group is nontrivial.

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group in which every endomorphism is trivial or an automorphism | any endomorphism is either the trivial map or an automorphism | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

directly indecomposable group | nontrivial and cannot be expressed as an internal direct product of nontrivial groups | |FULL LIST, MORE INFO | ||

freely indecomposable group | nontrivial and cannot be expressed as a free product of nontrivial groups | |FULL LIST, MORE INFO |

## Formalisms

### In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: retract

View other properties obtained by applying the simple group operator

The group property of being semidirectly indecomposable is obtained by applying the simple group operator to the subgroup property of being a retract (the simple group operator takes a subgroup property and outputs the property of being a group where there is no proper nontrivial subgroup having that property).

It is also obtained by applying the simple group operator to the subgroup property of being a complemented normal subgroup.