Update July 29, 2013.

I have still not found a good textbook for this topic, if you point one to me I will be grateful :) I have accepted BS's answer anyway, since their explanation was useful to me and gave me a good starting point for further research. I ended up finding a very helpful resource for this topic: the online notes of the course Introduction to Topological Groups, by Dikran Dikranjan, University of Udine. I recommend these notes to anyone interested on this topic, or on the Pontryagin-Van Kampen duality. The content of these notes has been partially published as An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups, Dikranjan and Stoyanov, Topology and its Applications, Volume 158, issue 15, 2011), p. 1942-1961, DOI: 10.1016/j.topol.2011.06.037, MR: 2825348, Elsevier Science.

A popular concept in quantum computation, used extensively to design algorithms for finite-abelian-groups, are the so-called orthogonal subgroups

Let $G=\mathbb{Z}_{d_1}\times\ldots\times\mathbb{Z}_{d_m}$ be a finite abelian group, the orthogonal subgroup $H^{\perp}$ of $H$ a subgroup of $G$ is defined as:

$$H^\perp:=\lbrace g\in H : \chi_g(h)=1 \quad\text{for all } h \in H\rbrace,$$

where $\chi_g$ are the characters of $G$: $$ \chi_g(h) = \exp{\left(2\pi \sum_{i=1}^{m} g_i h_i/d_i \right)} \quad \text{for all } \quad g, h \in G $$

Given two subgroups $H$ and $K$, basic Character Theory allows one to quickly derive

\begin{matrix} (1)\ H^{\perp\,\perp} = H & (2)\ |H^{\perp}| = |G|/|H| \\ (3)\ H\subset K \iff K^{\perp}\subset H^{\perp} & (4)\ (H\cap K)^{\perp} = \langle H^{\perp} , K^{\perp} \rangle \end{matrix}

Question.

This structure is extensively used in some important quantum algorithms and appears in quite a bunch of relatively-recent research papers. Yet, and though it looks pretty basic, I can not find some standard textbook where this is defined and that includes proofs of propositions (1-4). I would like to find such a reference since I often discuss these concepts with people not fluent with Character or Group theory. Also, I would like to know if the name "orthogonal-subgroup" is used by mathematicians.