Question:Normal subgroup multiple definitions

Q: '''There are so many different definitions of normal subgroup. Which of these is the correct definition? Which one should I use in practical situations?'''

A: The two most important definitions of normality are the definition as the kernel of a homomorphism of groups and the definition in terms of invariance under inner automorphisms. However, the other definitions are also important and worth knowing. What's really important about normality is not any one of these definitions per se, but the remarkable fact that all these definitions are equivalent. To see how different definitions can be used, both for proving that a subgroup is normal and for using that a subgroup is normal, see proving normality and using normality.