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Let $A$ be a maximal abelian (nilpotent) subgroup of the direct product $G=F\times H$ of groups $F$ and $H$.Then prove that $$A= (A\cap F)\times (A\cap H).$$

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The projections of $A$ onto $F$ and $H$ are both abelian/nilpotent. Hence so is the direct product of those projections, so by maximality $A$ is the direct product.

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