# nLab solvable group

A group is solvable if it is a finite iterated extension of an abelian group by abelian groups. In other words, there exists a finite sequence

$\{ 1\}\subset G_1 \subset G_2 \subset \ldots \subset G_k = G,$

in which $G_{j-1}$ is normal in $G_j$ and $G_j/G_{j-1}$ is abelian.

The terminology comes from elementary Galois theory: every polynomial equation $\phi$ over an integral domain $K$ has a corresponding Galois group $Gal(\phi/K)$, and $Gal(\phi/K)$ is a solvable group if and only if $\phi$ is a solvable equation (meaning that all its solutions in an algebraic closure of $K$ are expressible using the elements of $K$, the field operations, and extraction of roots).

Last revised on September 15, 2016 at 19:59:17. See the history of this page for a list of all contributions to it.