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 usually, 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]

↑ Boston, Nigel (2003), The Proof of Fermat's Last Theorem (PDF), Madison, Wisconsin, USA: University of Wisconsin Press

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