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.

See also

• Galois theory