# 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 has a corresponding Galois group, and the equation has a solution expressible using the field operations and extraction of roots if and only if its Galois group is a solvable group.

Revised on July 21, 2016 12:53:47 by Toby Bartels (64.89.52.46)