Let A,B be finite, commutative groups. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$, the set of homomorphisms from $A$ to $\mathbb{Q}/\mathbb{Z}$. $A^{*}$ is abelian itself (take this for granted). Let $f: B \to A$ a homomorphism of groups. For $h \in A^{*}$ define $f^{*}h \in B^{*}$ by $f^{*}h = h \circ f$. Take for granted that $f^{*}: A^{*} \to B^{*}$ is a homomorphism.
My questions are the following. I feel I am missing something really small that will enable me to prove both really easily. Any help will be appreciated:
a) $f$ injective $\iff$ $f^{*}$ surjective
b) Switch surjective and injective above.