Question:Normal subgroup abelian group relation

Q: '''The definition of normal subgroup seems to be somewhat related to the notion of abelian group. What precisely is the relation?'''

A: The main relation is abelian implies every subgroup is normal: in an abelian group, every subgroup is normal. This is best seen from the conjugation/inner automorphism definition, because in an abelian group, every element is invariant under conjugation by every other element. It can also be seen from the left coset/right coset definition or the commutator definition. Interestingly, there do exist non-abelian groups in which every subgroup is normal, such as the quaternion group.

It is not necessary that an abelian subgroup of a non-abelian group be normal. The easiest counterexample is the subgroups of order two in the symmetric group of degree three (see S2 is not normal in S3). It is also not true that any subgroup of an abelian normal subgroup is normal. An example is the dihedral group:D8, which has an abelian normal subgroups of order four (the Klein four-subgroups) which in turn have subgroups that are not normal in the whole group.