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Category of Nilpotent Groups

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Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements is a homomorphic image of N (i.e.: free objects do not always exist in the category of nilpotent groups).

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This provides a brief proof regarding a nilpotent group.

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Recall that the nilpotency class of a homomorphic image is bounded above by the nilpotency ...

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