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Let $f\colon A\to B$ be a group homomorphism between finite abelian groups.

For abelian group $G$, let $G^\wedge=\operatorname{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ be its Pontryagin dual.

Since $A,B$ are finite abelian groups, we have $A^\wedge=A, B^\wedge=B$.

Now, my question is :

How can we describe $f^\wedge\colon B\to A$ in terms of $f$?

Added: Can we relate $\operatorname{Ker} f^\wedge$, $\operatorname{Coker} f^\wedge$ with $\operatorname{Ker} f $, $\operatorname{Coker} f$?

1 Answers 1