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 $\text{Gal}(L/{\mathbf{𝐐}}_p)=(R⋊Q)⋊P$, 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