The title says it all, but let me repeat.


We all learn that the Pontryagin dual of a finite Abelian group is abstractly isomorphic as groups, but there’s no canonical isomorphism.


I think I understand it, but I don’t know how to formalize this statement, maybe using category theory.


Could someone enlighten me?


If many thinks that this is more suitable to Math.SE, please move this there.



How should I formalize that there’s no canonical isomorphism between a finite Abelian group and its Pontryagin dual?
Suppose we have a family of isomorphisms $\alpha\_A\colon A\to A^\*$ for all finite abelian groups $A^\*$. If $\phi\colon A\to B$ is an isomorphism, we have a dual isomorphism $\phi^\*\colon B^\*\to A^\*$ and thus an isomorphism $(\phi^\*)^{-1}\colon A^\*\to B^\*$. This makes $A^\*$ a covariant functor of $A$ on the category of finite abelian groups and isomorphisms, and it is only reasonable to call $\alpha$ canonical if it is natural with respect to this structure. In other words, we should have $(\phi^\*)^{-1}\circ\alpha\_A=\alpha\_B\circ\phi$ for all $\phi$, or $\alpha\_A=\phi^\*\circ\alpha\_B\circ\phi$. In particular, this must hold when $B=A$ and $\phi=n.1\_A$ for some $n$ that is coprime to the order of $A$. This means that $(n^2-1).\alpha\_A=0$, but $\alpha\_A$ is assumed to be an isomorphism, so the exponent of $A$ must divide $n^2-1$. This fails when $A=\mathbb{Z}/5$ and $n=2$, for example, so there is no natural map $\alpha$ as described.


The same argument also shows that there is no natural isomorphism $V\to V^\*$ for finite-dimensional vector spaces $V$ over $\mathbb{Z}/p$ provided that $p>3$. The same conclusion holds for $p=2$ or $p=3$ but one needs to use some different choices of $\phi$ to prove it.


On the other hand, if we restrict even further to elementary abelian groups of order $4$, then you can check that there is a natural choice of $\alpha\_A$. It sends each nonzero element $a\in A$ to the unique map $\theta\colon A\to\mathbb{Z}/2$ such that $\theta\neq 0$ but $\theta(a)=0$. 


