Hello

Take a group, G. then N is called a normal subgroup of G iff and ,

You can also say N is a normal subgroup of G if the sets of left and right cosets of N in G coincide, or ,

Say H is a subgroup of an abelian group, G.also, is a subgroup of an abelian group, necessarily abelian?

Take

Then since H is a subgroup,

Since G is abelian, you can say

So by closure

So yes, a subgroup of an abelian group is abelian.