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 usually, 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.^[1]