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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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  • $\begingroup$ Could you perhaps move the short proof to an appendix? $\endgroup$
    – Yemon Choi
    Commented Jan 18, 2012 at 1:14
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    $\begingroup$ (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. $\endgroup$
    – Yemon Choi
    Commented Jan 18, 2012 at 1:15
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    $\begingroup$ @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. $\endgroup$
    – Carl
    Commented Jan 18, 2012 at 1:19

3 Answers 3

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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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There are very pedagogical proofs in [1] section 3.5 and in [2] section 6.1. In both references, the most typical properties of orthogonal subgroups of finite-abelian groups are proven, including the one you are looking for. I have read both texts and used both for preparing seminars, and I find the exposition quite clear. I have cited both of these references in one of my papers: I think the first of them [1] has never appeared in a Journal, but it is quite a trusted paper in my field (quantum computation), so you could cite the arXiv; the second reference [2] has recently been published in QIC Vol. 13. No.11&12, 1007 (2013).

In fact, this property is one of many consequences of the Pontryagin-Van Kampen duality, and holds for all locally compact Abelian groups (finite Abelian groups are a particular case of those; the integers, the real numbers, the torus are also locally compact Abelian; see also this related question of mine in MathOverflow). If you would like to see a proof of your statement in the more general case, I recommend you to take a look at the online notes of the course Introduction to Topological Groups, by Dikran Dikranjan, University of Udine. The relevant sections for you are:

  • Section 11 Pontryagin-van Kampen duality, in particular 11.4.;
  • 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;
  • theorem 11.5.4. proves your statement.

As far as I know, Dikranjan's notes have never been published; still, they are (in my opinion) a very useful resource. Some parts of them (but sadly not the ones you need) have appeared in An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups, Dikranjan and Stoyanov, Topology and its Applications, Vol. 158, no. 15, 2011 Elsevier Science.

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