Soluble group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

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} = G₀ < G₁ < ⋅⋅⋅ < G_k = G such that G_{j − 1} is normal in G_j, and G_j/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 {1} of G. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient  ^H⁄_N is abelian if and only if N includes H⁽¹⁾. The least n such that G⁽ⁿ⁾ = {1} is called the derived length of the solvable group G.

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