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}G 1G 2G k=G, \{ 1\}\subset G_1 \subset G_2 \subset \ldots \subset G_k = G,

in which G j1G_{j-1} is normal in G jG_j and G j/G j1G_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)