Normal closure (group theory)

In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing : The normal closure is the smallest normal subgroup of containing in the sense that is a subset of every normal subgroup of that contains The subgroup is generated by the set of all conjugates of elements of in Therefore one can also write Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set is the trivial subgroup.