# Dual concept for the p-primary component

Is there a dual concept for the p-primary component of an abelian group? Please name some books/papers where it is studied.

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Posted at the same time to M.SE: math.stackexchange.com/questions/311702/… This practice is frowned upon! – Steve D Feb 23 '13 at 5:12

## 1 Answer

EDIT: Note that in what follows I interpret "dual" as "dual in the sense of Pontryagin-Van Kampen's duality". If you had something different in mind I think you should have specified...

Well, first of all you should start finding a dual notion for the torsion part. Indeed, a given Abelian discrete Abelian group $G$ is the central element in a short exact sequence: $$0\to t(G)\to G\to G/t(G)\to 0 ,$$ where $t(G)$ is the torsion part and $G/t(G)$ is torsion free. As the duality functor (in the Pontryagin-Van Kampen duality) is exact you get a short exact sequence of compact Abelian groups: $$0\to \widehat{G/t(G)}\to \widehat G\to \widehat{t(G)}\to 0 .$$ Recall also that the torsion part $t(G)$ can be definite to be the direct union of all the finite subgroups of $G$. Using that finite Abelian groups are self-dual and that the duality functor send direct to inverse limits, you obtain that $\widehat{t(G)}$ is an inverse limit of finite Abelian groups, that is, it is pro-finite. In fact, it is the maximal pro-finite quotient of $\widehat G$.

Now, coming to your question, you have to recall that $t(G)$ is the direct sum of its $p$-primary components $t_p(G)$ (each one defined as the direct union of all the finite $p$-subgroups of $G$) and that the duality functor sends direct sums (i.e., coproducts) to products. Thus $\widehat {t(G)}$ is the (topological) product of the $\widehat{t_p(G)}$ and each $\widehat{t_p(G)}$ is the inverse limit of $p$-torsion finite groups, that is, pro $p$-finite.

To conclude, the answer to your question is: the dual of the $p$-torsion subgroup is the maximal pro $p$-finite quotient of the dual group.

As you are asking for references, I think that any standard book with an exposition of the Pontryagin-Van Kampen's duality, if not the above discussion, will certainly give you proofs of all the facts I used in my answer. I personally like Dikran Dikranjan's approach to the duality theorem: http://users.dimi.uniud.it/~dikran.dikranjan/ITG.pdf

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