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.[1]

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