**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][1]*, 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, Elsevier Science.

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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 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 use 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.


  [1]: http://users.dimi.uniud.it/~dikran.dikranjan/ITG.pdf