# Directly indecomposable 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 **directly indecomposable** if it satisfies the following equivalent conditions:

- It has no proper nontrivial direct factor
- It cannot be expressed as an internal direct product of nontrivial subgroups
- It is not isomorphic to the external direct product of any two nontrivial groups

## Formalisms

### In terms of the simple group operator

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

View other properties obtained by applying the simple group operator

The group property of being directly indecomposable is obtained by applying the simple group operator to the subgroup property of being a direct factor.

## 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 normal subgroup | Group having no proper nontrivial transitively normal subgroup, Monolithic group, Splitting-simple group, Subdirectly irreducible group|FULL LIST, MORE INFO | ||

subdirectly irreducible group | nontrivial and cannot be expressed as a nontrivial subdirect product | |FULL LIST, MORE INFO | ||

splitting-simple group | nontrivial and no proper nontrivial complemented normal subgroup | |FULL LIST, MORE INFO | ||

centrally indecomposable group | nontrivial and any central factor is inside the center | |FULL LIST, MORE INFO | ||

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

almost simple group | has a normal fully normalized subgroup that is simple non-abelian | |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 | Splitting-simple group|FULL LIST, MORE INFO |

## Facts

### Products of directly indecomposable groups

It is clear that every finite group can be expressed as a product of directly indecomposable groups. The question: is this expression as a product essentially unique? That is, is there an analogue of unique factorization for direct products? The answer is yes, as per the Remak-Schmidt theorem.