A normal subgroup of an abelian group is a subgroup that is invariant under conjugation by any element of the group. In other words, if a subgroup is normal, then it remains unchanged when the elements of the group are multiplied by any element in the group.
To prove that a subgroup is normal in an abelian group, you must show that for any element in the subgroup and any element in the group, the element's conjugate is also in the subgroup. This can be done by using the definition of a normal subgroup and the properties of an abelian group.
No, a subgroup of an abelian group cannot be both normal and non-normal. If a subgroup is normal, then it must be invariant under conjugation by any element of the group. If it is not normal, then there exists at least one element in the group whose conjugate is not in the subgroup.
Proving that a subgroup is normal in an abelian group allows us to better understand the structure and behavior of the group. It also allows us to make certain conclusions about the group based on the properties of normal subgroups, such as the quotient group theorem.
Yes, there are other methods for proving normality of subgroups in abelian groups, such as using the first isomorphism theorem or showing that the subgroup is a kernel of a homomorphism. However, the most direct and commonly used method is to show that the subgroup is invariant under conjugation.