 It has associated field extensionsformula_12giving a solvable group containing formula_13 (acting on the formula_14) and formula_15 (acting on formula_16).A group "G" is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = "G"0 < "G"1 < ⋅⋅⋅ < "Gk" = "G" such that "G""j"−1 is normal in "Gj", and "Gj "/"G""j"−1 is an abelian group, for "j" = 1, 2, …, "k".Or equivalently, if its derived series, the descending normal series
where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of "G".