Hello! Given a surjective group homomorphism . Why is the restriction to the commutator subgroup also surjective?
Greetings
Banach
Quite simply, because it's a homomorphism.
Let . Then, and are homomorphic images of elements from , because the mapping is onto. So substitute in these elements, , and then use the fact that is a homomorphism to see that it is in the image of . Thus the generators of are contained in the image, and so the whole subgroup is.
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