Unfactorizable group

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.

Weaker properties

 * Directly indecomposable group
 * Semidirectly indecomposable group