Editing Subgroup series (section)

===Solvable and nilpotent===
* A '''[[solvable group]]''', or soluble group, is one with a subnormal series whose factor groups are all [[Abelian group|abelian]].
* A '''[[nilpotent series]]''' is a subnormal series such that successive quotients are [[nilpotent group|nilpotent]].
:A nilpotent series exists if and only if the group is [[solvable group|solvable]].
* A '''[[central series]]''' is a subnormal series such that successive quotients are [[center (group)|central]], i.e. given the above series, <math>A_{i+1}/A_i \subseteq Z(G/A_i)</math> for <math>i=0, 1, \ldots, n-2</math>.
:A central series exists if and only if the group is [[nilpotent group|nilpotent]].