Prove that a x b is an abelian group if and only if both

Question

Q1: Prove that if x² =1 for all x ∈ G  then G is abelian.

Q2: Prove that A x B is an abelian group if and only if both and are abelian.

Q3: Let be an abelian group and fix some n ∈ z. Prove that the set {aⁿ|a ∈ A} is a subgroup of A.

Q4: Let H ≤ G and define a relation ~ on G by a~b if and only if b^-1a ∈ H.

Prove that ~ is an equivalence relation.

Q5: Prove that if H and K are finite subgroups of G whose orders are relatively prime then H ∩ K = 1.