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 $𝐐_{p}$ . Then the Galois group $Gal ({\overline{𝐐}}_{p} / 𝐐_{p})$ , where ${\overline{𝐐}}_{p}$ denotes the algebraic closure of $𝐐_{p}$ , is prosolvable. This follows from the fact that, for any finite Galois extension $L$ of $𝐐_{p}$ , the Galois group $Gal (L / 𝐐_{p})$ can be written as semidirect product $Gal (L / 𝐐_{p}) = (R ⋊ Q) ⋊ P$ , with $P$ cyclic of order $f$ for some $f \in 𝐍$ , $Q$ cyclic of order dividing $p^{f} - 1$ , and $R$ of $p$ -power order. Therefore, $Gal (L / 𝐐_{p})$ is solvable.^[1]