Group Theory

1. Let K,N be normal subgroups of a group G. Suppose that the quotient groups G/K and G/N 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 H of a group G, H is normal in G and G/H is an abelian group if and only if the commutator subgroup D(G)=[G, G] is contained in H.

Using this lemma, we know that G/K is an abelian group if and only if the commutator subgroup D(G)=[G, G] is contained in K.

Similarly, since G/N is abelian, D(G)is contained in N.
Therefore, the commutator subgroup D(G)⊂K∩N. This implies, again by Lemma, that the quotient group G/(K∩N) is an abelian group as required.