I came across an exercise on Abelian groups which said that,

"In an abelian group G, every subgroup is normal". I had no issues proving this result. But my question is this.

Re: Every subgroup of an abelian group G is normal.

# 1. Does this mean that the group G, an improper subgroup of itself, is also normal?

# 2. Does this mean that commutativity implies normality of groups?

Is this similar to saying that every cyclic group is abelian?

# 3. If 2 is not true, are there any specific conditions under which 2 is true or false?