Final answer:
The smallest normal subgroup of a group G containing a subset S is well-defined and can be constructed by taking the intersection of all normal subgroups of G that contain S, referred to as the normal closure of S in G.
Step-by-step explanation:
The student is asking about the concept of a normal subgroup in group theory, a branch of abstract algebra in mathematics. Specifically, the student wants to understand how to construct the smallest normal subgroup of a group G that contains a given subset S.
This is a well-defined concept because one can create such a subgroup by taking the intersection of all normal subgroups of G that contain S. This intersection itself is a normal subgroup of G (since the intersection of normal subgroups is also normal) and it contains S.
Additionally, it is the smallest such subgroup since any normal subgroup of G that contains S must also contain this intersection. Thus, this intersection is called the normal closure of S in G.