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This question arises from reading the note Hodge cycles on abelian varieties by P. Deligne (notes by J.S. Milne). Suppose we are given a group $G$ (for example, either a fundamental group $\pi_1(S, s_0)$ or a Galois group $Gal(k/k_0)$) acting on a finite dimensional $\mathbb{Q}$-vectors space. It seems to me that the author concludes that the action factors through a finite subgroup once it is known there is a symmetric positive definite $\mathbb{Q}$-valued bilinear form on $V$. Of course, this is note true in general (Take $V=\mathbb{Q}^2$ with the standard inner product, for example.) So I am wondering what is the missing link in my understanding of the proofs. The proofs in question are Theorem 2.15 and (iii) of Propositin 2.9(b). Why does the action factors through finite subgroups in those proofs?

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In the first case, the image of the group is obviously a discrete subgroup of $Aut(V\otimes \mathbb{R})$, hence discrete and compact. In the second case, the group preserves a lattice in $V$, and so the image is again discrete and compact.

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