# Unfactorizable 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 simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

## Contents

## Definition

### Symbol-free definition

A group is said to be **unfactorizable** if it satisfies the following equivalent conditions:

- It has no proper nontrivial permutably complemented subgroup
- It does not possess an exact factorization (viz it cannot be written as the product of a matched pair of subgroups)

### Definition with symbols

A group is said to be unfactorizable if for any choice of subgroups and such that is trivial and , and are (in some order) the whole group and the trivial subgroup.

### In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: permutably complemented subgroup

View other properties obtained by applying the simple group operator