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 ϕ\phi over an integral domain KK has a corresponding Galois group Gal(ϕ/K)Gal(\phi/K), and Gal(ϕ/K)Gal(\phi/K) is a solvable group if and only if ϕ\phi is a solvable equation (meaning that all its solutions in an algebraic closure of KK are expressible using the elements of KK, the field operations, and extraction of roots).

Revised on September 15, 2016 19:59:17 by Dexter Chua (