If the quotients of a group by two subgroups are abelian, then the quotient by their intersection is abelian

Let A,B \leq G be normal subgroups such that G/A and G/B are abelian. Prove that G/(A \cap B) is abelian.


Since G/A is abelian we have [G,G] \leq A. Likewise, [G,G] \leq B. Thus [G,G] \leq A \cap B, so that G/(A \cap B) is abelian.

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