# Unfactorizable group

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

## 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 $G$ is said to be unfactorizable if for any choice of subgroups $H$ and $K$ such that $H \cap K$ is trivial and $HK= G$, $H$ and $K$ 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