Algorithmic questions for linear algebraic groups. I

Let $G$ be a linear algebraic group defined over the field of rational numbers and subject to certain conditions, let $G(\mathbf R)$ be its group of real points, and let $G(\mathbf Z,m)$ be a congruence-subgroup of its group of integer points. In this paper it is proved that, using a recursive procedure, one can construct a fundamental set of $G(\mathbf Z,m)$ in $G(\mathbf R)$. This result will be applied in the second part of the article.
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