**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.

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