Prosolvable group

In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group.

Examples

 * Let p be a prime, and denote the field of p-adic numbers, as usual, by $$\mathbf{Q}_p$$. Then the Galois group $$\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$$, where $$\overline{\mathbf{Q}}_p$$ denotes the algebraic closure of $$\mathbf{Q}_p$$, is prosolvable. This follows from the fact that, for any finite Galois extension $$L$$ of $$\mathbf{Q}_p$$, the Galois group $$\text{Gal}(L/\mathbf{Q}_p)$$ can be written as semidirect product $$\text{Gal}(L/\mathbf{Q}_p)=(R \rtimes Q) \rtimes P$$, with $$P$$ cyclic of order $$f$$ for some $$f\in\mathbf{N}$$, $$Q$$ cyclic of order dividing $$p^f-1$$, and $$R$$ of $$p$$-power order. Therefore, $$\text{Gal}(L/\mathbf{Q}_p)$$ is solvable.