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Let $G$ be a finite abelian group, and denote by $G^{\times}$ its set of linear characters, i.e. homomorphisms $\phi : G \to \mathbb C^{\times}$, where $\mathbb C^{\times}$ denotes the multplicative group of the complex numbers. Then these form an orthonormal set with respect to the inner product $$ \langle \chi, \phi \rangle := \frac{1}{|G|} \sum_{g\in G} \chi(g) \overline{ \phi(g)}. $$ In these lecture notes by Daniel Bump it is proven that $|G^{\times}| \le |G|$, this is part of the proof of Proposition 2.2.3, namelely that $|G| = |G^{\times}|$ and that $G \cong (G^{\times})^{\times}$. The argument goes like this:

We first observe that $|G^{\times}| \le |G|$ since the linear characters are an orthonormal set, hence lineary independent.

I do not get it, what is the connection between linear independence of the characters and the cardinality of $|G|$? Could you please clarify?

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    $\begingroup$ Characters live in the vector space of all functions $G \to \mathbb{C}$, which has dimension $|G|$. $\endgroup$ – Qiaochu Yuan Oct 8 '15 at 20:30
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    $\begingroup$ Hope they have a life worth living there, thanks for your comment! :) $\endgroup$ – StefanH Oct 8 '15 at 20:36

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