On the decomposition of automorphisms of free modules into simple factors

摘要

We study decompositions of automorphisms of various free modules into products of transvections and dilations. In particular, for a free Z-moduIe M = Z" (where n > 3) we show that any automorphism a GLnM) can be expressed as a product of not more than In + 5 transvections and one simple transformation which is a trans vection if a e SLn(M), and a dilation otherwise. As a corollary we obtain that for n > 3*the width of the group SLn(Z), with respect to the set of commutators, does not exceed 10.