Normal Subgroup Determination in Z6

Let G be $\mathbb{Z}_6$ and H be the set {0, 3}. Are the left cosets and the right cosets the same, making H a normal subgroup of G?

In this problem, we explore the properties of cosets in the context of group theory, particularly examining whether the subgroup H of G is normal. Understanding cosets is crucial for delving into the deeper structures of groups, as cosets partition the group into disjoint subsets.

The group G is given as $\mathbb{Z}$ mod 6, which is a cyclic group comprising the integers modulo 6. Subgroups and cosets are fundamental in understanding how groups function and are structured. H being a subset containing the elements 0 and 3 indicates its potential to be a subgroup of G, satisfying the key group properties.

The key concept here is to check whether the left cosets and the right cosets formed by H are the same, as this equality implies normality of a subgroup. This scenario isn't just about calculating cosets but also involves an understanding of how normal subgroups interact within the larger group structure and how they affect group operations. A normal subgroup has the property that it is invariant under conjugation by any element of the group, a crucial concept when dealing with quotient groups and further applications in algebra.