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.