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