**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](http://users.dimi.uniud.it/~dikran.dikranjan/ITG.pdf)*, 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](http://www.sciencedirect.com/science/article/pii/S0166864111002604)*, Dikranjan and Stoyanov, Topology and its Applications, Volume 158, issue 15, 2011), p. 1942-1961, DOI: [10.1016/j.topol.2011.06.037](https://dx.doi.org/10.1016/j.topol.2011.06.037), MR: [2825348](http://www.ams.org/mathscinet-getitem?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.



Finite, abelian, yet "fugitive" orthogonal subgroups
These results belong to what is called *duality in finite abelian groups*, a theory that has been generalized by [Pontryagin](http://en.wikipedia.org/wiki/Pontryagin_duality) and others in the 30's to locally compact abelian groups. 


Another keyword here is "Discrete Fourier Transform", although it is mainly used for cyclic groups (of order $2^N$ for FFT) or $GF(2)$ vector spaces (analysis of boolean functions).


It is also a chapter in the representation theory of finite groups, founded by Frobenius in the end of 19th century, namely the case of finite abelian groups, where the irreducible characters are all homomorphisms to $S^1$. This case was already known to Gauss and Dirichlet. 


By the way, the orthogonal you define is more naturally seen as a subgroup of the *dual group* $\hat{G}=Hom(G,S^1)$, which in (multiplicative) duality with $G$ via the evaluation $G\times \hat{G}\to S^1$.
It is isomorphic to $G$, but not naturally so. 


I would bet that almost any book on either Pontryagin duality or representation theory of finite groups has the results you want to cite, but I have only been able to find [this link](http://www.encyclopediaofmath.org/index.php/Pontryagin_duality) to an online encyclopedy, which easily implies your claims, since 3 and 4 follow quickly from 1 and 2. 


Hope this helps.


