Splitting-simple group

Symbol-free 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

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.

Stronger properties

 * Weaker than::Simple group
 * Weaker than::Group in which every endomorphism is trivial or an automorphism:

Weaker properties

 * Stronger than::Directly indecomposable group
 * Stronger than::Freely indecomposable group