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Does every normal subgroup equal to the normal closure of some set of a group?
A normal subgroup is a subgroup of a group that satisfies the property that for any element in the normal subgroup and any element in the original group, the product of the two elements and the inverse of the first element must also be in the normal subgroup.
Normal subgroup equality is closely related to closure of a group. In fact, a group is considered closed if and only if it contains all of its normal subgroups.
Normal subgroup equality helps us to better understand the structure and properties of a group. It also allows us to make connections between different groups and their subgroups.
Normal subgroup equality is a more specific concept than subgroup equality. While subgroup equality simply requires that a subgroup is a subset of the original group, normal subgroup equality also requires that the subgroup satisfies the property mentioned in the first question.
Yes, a group can have multiple normal subgroups. In fact, every group contains at least two normal subgroups: itself and the trivial subgroup consisting of just the identity element.