Is every abelian group normal?

I'm curious about a fundamental concept in group theory. Can you clarify for me: is it true that every abelian group is necessarily normal? It seems that abelian groups possess a certain level of symmetry and commutativity, which might suggest they inherently possess the properties of normality. However, I'm unsure if this is always the case. Could you elaborate on the relationship between abelian groups and normality, and if there are any exceptions or nuances to this potential connection?