The solvable length of groups of local diffeomorphisms

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we show that {2n+1} is the sharpest upper bound for the derived length of solvable subgroups of the group {\mathrm{Diff}({\mathbb{C}}^{n},0)} of local complex analytic diffeomorphisms for {n=2,3,4,5} .

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group [Formula: see text] into conjugate subgroups, [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of [Formula: see text]. We also show that every solvable group [Formula: see text] is a product of at most [Formula: see text] conjugates of a Carter subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.