Is every solvable subgroup of \$GL(n,\mathbb{Z})\$ polycyclic? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:35:52Z http://mathoverflow.net/feeds/question/79067 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79067/is-every-solvable-subgroup-of-gln-mathbbz-polycyclic Is every solvable subgroup of \$GL(n,\mathbb{Z})\$ polycyclic? HenrikRüping 2011-10-25T12:12:59Z 2011-10-25T21:10:42Z <p>Is every solvable subgroup of \$GL(n,\mathbb{Z})\$ polycyclic?</p> <p>The first solvable group that is not polycyclic is \$\mathbb{Z}[1/2]\rtimes \mathbb{Z}\$ (where the automorphism is given by multiplication with 2) and I do not see a way of embedding it into \$GL_n(\mathbb{Z})\$ for some \$n\$. </p> http://mathoverflow.net/questions/79067/is-every-solvable-subgroup-of-gln-mathbbz-polycyclic/79071#79071 Answer by Gjergji Zaimi for Is every solvable subgroup of \$GL(n,\mathbb{Z})\$ polycyclic? Gjergji Zaimi 2011-10-25T12:36:35Z 2011-10-25T21:10:42Z <p>It is a theorem of Mal'cev that all solvable subgroups of \$GL(n,\mathbb Z)\$ are polycyclic, and a theorem of Auslander that every polycyclic group is isomorphically embeddable in \$GL(n,\mathbb Z)\$, for some \$n\$. Auslander's theorem was later reproved by Swan purely algebraically by adapting the proof of Ado's theorem.</p> <blockquote> <p>A. I. Mal’cev, "On certain classes of infinite solvable groups", Mat. Sb. 28 (1951) 567–588; Amer. Math. Soc. Transl. (2) 2 (1956) 1–21</p> <p>R.G. Swan, "Representations of polycyclic groups", Proc. Amer. Math. Soc, 1967</p> <p>L. Auslander, "On a problem of Philip Hall", The Annals of Mathematics, 1967 </p> </blockquote>