yes

isn't this just rephrasing the same question? if a group is commutative, it is Abelian, and if it's Abelian, all its subgroups are normal.# 2. Does this mean that commutativity implies normality of groups?

not really. here we are talking about the group itself. but the theorem in question goes beyond that. not only do we know something about the group itself, but every subgroup of that group. i suppose the universal quantifier makes it similar. the "every" part.Is this similar to saying that every cyclic group is abelian?