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Supposing that the simple abelian subvarieties of A are pairwise non-isogenous, we find a lower bound for the rank of G, which is a little less than log_2 dim A. If we suppose that End A is commutative, then we show that rk G >= log_2 dim A + 2, and this latter bound is sharp. We also obtain the same results for the rank of the l-adic monodromy group of an abelian variety defined over a number field.\n  -----\n  Soit A une vari\\'et\\'e ab\\'elienne complexe et G son groupe de Mumford--Tate. 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