Proposition 8

— Suppose that G and H are commutative and annihilated by n. If sf is bijective and H is finite, then d_f is bijective and we have Card(G) = Card(H).*

If s_f is bijective and H is finite, we have by Cor. 2 to Prop. 7 the relation Card(G) = Card(Hom(H, T)) = Card(H), hence Card(G) is finite and by another application of the Corollary, Card(Hom(G, T)) = Card(H). So it is enough to prove that d_f is injective. Now if h \in \mathrm{Ker}(d_f), we have f(g, h) = 1 for all g \in G, hence since s_f is bijective, \varphi(h) = 1 for all \varphi \in \mathrm{Hom}(H, T); by Prop. 7 this implies \mathrm{Hom}(\mathrm{Ker}(d_f), T) = \{1\}, hence \mathrm{Ker}(d_f) = \{1\} by Cor. 1 to Prop. 7.