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?
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.