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]

See also

• Galois theory

References

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