Hi All. I face a following problem:
Let i have theorem 1: There is an integer-valued function $f_s(n)$ of the positive integer n only such that for any division ring D of characteristic zero and any finite subgroup G of GL(n,D) with $C_G(N)\leq N$, for N the Fitting subgroup of G, the group G contains a metabelian normal subgroup of index at most $f_s(n)$. If G is a finite soluble group then always $C_G(N)\leq N$.
How can i infer this result: ]). If G is locally soluble then G has a metabelian normal subgroup of finite index at most $f_s(n)$, the function of theorem 1. In particular G is soluble with bounded derived length.
I really appreciate your all ideas! Thanks!