Let G be a group. H is a subgroup of G. If G/H is abelian and H is abelian, Is G abelian? I think not. It is given Hxy = Hyx This just implies xy(x^-1)(y^-1) is in H So does this mean H is a commutator subgp? And is G abelian?
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Try $\displaystyle G=S_3,\ H=A_3.$
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