Nearly normal subgroup

Symbol-free definition
A subgroup of a group is said to be nearly normal if it satisfies the following equivalent conditions:


 * 1) It has finite index in its defining ingredient::normal closure.
 * 2) It is a subgroup of finite index of a normal subgroup of the whole group.

Related properties

 * Almost normal subgroup

Facts

 * Every subgroup of a group is nearly normal if and only if the derived subgroup is finite, viz., the group is a commutator-finite group.