# Orthogonal subgroups of dual group

This question arises from this one.

Let $G$ be a finite abelian group and $H$ a subgroup of $G$. Let $\widehat{G}$ be group of all characters of $G$ and let $H^\perp = \{\chi \in \widehat{G} : \chi = 1 \text{ on } H\}$. Does anyone know a good reference proving the identity $(H^\perp)^\perp = H$?

I'm interested in a reference (instead of a proof) because I only need to use this fact in a non representation theory paper and won't have the space needed to introduce and prove what is needed.

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Could you perhaps move the short proof to an appendix? –  Yemon Choi Jan 18 '12 at 1:14
(you just need to set up the definitions, and then observe that the canonical map from $H$ into $(H^\perp)^\perp$ is injective; finiteness then implies surjectivity. –  Yemon Choi Jan 18 '12 at 1:15
@Yemon Choi: I guess it could be an option. It's a theoretical computer science paper, so I thought I could avoid doing so because it'd be standard in representation theory and somewhere in some well known textbook. –  Carl Jan 18 '12 at 1:19

Lemma 2.1.3 in Rudin's Fourier Analysis on Groups does this for locally compact abelian $G$ and closed $H \leq G$. This might not the best reference (may be a bit too general for a CS readership?), but it was the first one to pop into my head.

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It is Proposition 3.4 in the book "Washington: Introduction to Cyclotomic Fields".

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• the annihilator $A_{\widehat{G}}(H)$, defined at the beginning of section 11.4.2, is precisely the subgroup of the characters of $G$ that you are looking at;