1. Let be normal subgroups of a group . Suppose that the quotient groups and are both abelian groups.
Then show that the group G/(K∩N) is also an abelian group.
Answer: We use the following fact to prove the problem.
Lemma: For a subgroup of a group , is normal in and is an abelian group if and only if the commutator subgroup is contained in .
Using this lemma, we know that is an abelian group if and only if the commutator subgroup is contained in .
Similarly, since is abelian, is contained in .
Therefore, the commutator subgroup G/(K∩N) is an abelian group as required.. This implies, again by Lemma, that the quotient group