Uncertainty in the definition of "Dual Group"? Referring to Lang's Algebra p. 46, given an abelian group $G$ of exponent $m$, the dual group is defined to be $Hom(G,Z_m)$ and is denoted by $G^{\wedge}$. This is were it does not feel right: every multiple of $m$ is an exponent as well and so we could have defined $G^{\wedge}$ as $Hom(G,Z_{km})$ for any integer $k>1$. Does this not mean that this definition is not well-posed? On the contrary if we were to restrict $m$ to be the smallest exponent of $G$, then we have a unique dual group $G^{\wedge}$. Any insights?
 A: Let $m$ be the smallest exponent of $G$ (or if you're not Lang the exponent of $G$). Then every element of $G$ has order dividing $m.$ The same is true of image of $G$ under a homomorphism. Therefore, under any homomorphism of $G$ into a $\mathbb{Z}/km,$ it must be the case that the image of $G$ lies in the unique subgroup of $\mathbb{Z}/km$ of order $m.$ Consequently,
$$Hom(G,\mathbb{Z}/km) = Hom(G,\mathbb{Z}/m)$$
and there is no problem with a wellposedness. 
Observing this, we can make a definition for the dual group that does not mention the exponent of the group.
Definition: Given a finite Abelian group $G,$ the dual group $G^{\wedge}$ is the group of homomorphisms from $G$ into the colimit of the diagram of all finite cyclic groups i.e.
$$G^{\wedge} := Hom(G,\mathbb{Q}/\mathbb{Z})$$
One easily checks that this definition coincides with the one given by Lang.
A: If you want to have a reasonable theory, the framework is the study of the category of (not necessarily finite)  $\mathbb Z/n \mathbb Z$-modules.
That category is isomorphic to the full subcategory of abelian groups obtained by taking as objects the groups  $A$ killed by $n$ i.e. such that $nA=0$.
The important point is to fix $n$ and never to worry about the (exact) exponent of $A$,  which is essentially irrelevant or, worse, error inducing.
Fixing a cyclic group $T$ of order $n$, you consider the group morphisms $Hom(A,T)$and obtain an easy duality "theory". 
The best known application  is Kummer theory, the chapter of Galois theory studying abelian field extensions whose Galois groups are precisely  those  abelian groups killed by $n$.
Said differently, Kummer theory  studies extensions obtained by starting with  a field $K$ containing a primitive $n$-th root of unity, taking  a bunch of equations $X^n-a=0$ with $a\in K$ and adjoining their zeros to $K$.
Irrelevant remark I mentioned an isomorphism of categories in my second sentence, and not just an equivalence. In my practice of  mathematics this doesn't happen very often.
