#then there’s an obvious and sensible way to construct ψ : G/N -> G’/N’ such that everything commutes
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Consider now the Cainian functor C : Grp → Grp given by G ↦ [G, G], which annihilates all abelian groups,
#math#I did not actually check that this is a functor…but it feels like it should be#but also that it might not#EDIT: oh yeah obviously a functor! given N and N’ normal subgroups of G and G’ respectively#as long as ɸ : G -> G’ restricted to N lands in N’#then there’s an obvious and sensible way to construct ψ : G/N -> G’/N’ such that everything commutes#this is probably one of those real basic theorems but it’s been a while and I’ve just got to rederive everything…#EDIT 2: i initially had G ↦ G/Z(G)#but it was pointed out in the replies that the center doesn’t necessarily land in the center so the above doesn’t apply!#however this is obviously a functor.#i say having learned nothing about declaring things ‘obvious’.
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