Given a group and a subset , the normal closure of in is the smallest subgroup containing the set of all conjugates of elements in , i.e. the subgroup generated by the set of all elements of the form , where and . The normal closure is clearly a normal subgroup of .
The normal closure, also called conjugate closure (see Wikipedia), should be distinguished from the normalizer of in . In combinatorial group theory a group is presented via specifying a set of generators and a set of relations. This means that where is the free group generated by and is the normal closure of in .