Complemented central factor

Definition
A subgroup of a group is termed a complemented central factor or split central factor if it satisfies the following equivalent conditions:


 * 1) It is a permutably complemented subgroup as well as a central factor of the whole group.
 * 2) It is a defining ingredient::complemented normal subgroup as well as a central factor of the whole group.

Metaproperties
Both the whole group and the trivial subgroup are complemented central factors.

If $$H$$ is a complemented central factor of a group $$G$$, then $$H$$ is also a complemented central factor in any intermediate subgroup $$K$$. This follows because both the property of being a permutably complemented subgroup and the property of being a central factor satisfy the intermediate subgroup condition.