When is Out$(SL_n(R))$ a torsion group ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:23:20Z http://mathoverflow.net/feeds/question/59822 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59822/when-is-outsl-nr-a-torsion-group When is Out$(SL_n(R))$ a torsion group ? HenrikRüping 2011-03-28T12:17:47Z 2011-04-08T03:10:16Z <p>This question is a follow up question to <a href="http://mathoverflow.net/questions/57235/automorphisms-of-sl-n-mathbbz/57236#57236" rel="nofollow">this</a> question. So my question is:</p> <p>For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of Theorem A and B in <a href="http://www.reference-global.com/doi/abs/10.1515/crll.1966.223.56" rel="nofollow">O'Meara The automorphisms of the linear groups over any integral domain</a> is that this is the case (for $n\ge 3$) for any integral domain, whose underlying additive abelian group is finitely generated. </p> <p>However this is just a computation and I am wondering, whether this question has already been studied somewhere more systematically or if there are other results that also have such a corollary.</p> http://mathoverflow.net/questions/59822/when-is-outsl-nr-a-torsion-group/61017#61017 Answer by Mark Sapir for When is Out$(SL_n(R))$ a torsion group ? Mark Sapir 2011-04-08T02:10:05Z 2011-04-08T03:10:16Z <p>This question has been studied even for non-commutative associative rings. See, for example Golubchik, I. Z.; Mikhalëv, A. V. Isomorphisms of the general linear group over an associative ring. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 3, 61–72. They prove that if the ring $R$ contains $1/2$ (that is $2$ is invertible), then every isomorphism $\phi: GL_n(R)\to GL_n(R)$, $n\ge 3$, is standard on the subgroup $GE_n(R)$ generated by the elementary and diagonal matrices. "Standard" means "generated by an automorphism of $R$. </p>