Nonsolvable

Nonsolvable is an adjective used in mathematics, typically in group theory, to describe a group that is not solvable.

In group theory, a group G is solvable if there exists a finite chain of subgroups {1} = G0 ⊲ G1 ⊲ ... ⊲ Gr = G such that each quotient Gi+1/Gi is abelian. Equivalently, G is solvable if its derived series G^(0) = G, G^(i+1) = [G^(i), G^(i)] eventually reaches the trivial group after finitely many steps. A group that does not satisfy these conditions is nonsolvable.

For finite groups, solvability can also be described by the composition series: a group is solvable iff all its composition factors are cyclic of prime order. In particular, the presence of a non-abelian simple factor implies nonsolvability.

Smallest example and standard cases: A5, the alternating group on five elements, is the smallest nonsolvable group. It is simple and non-abelian, so its derived subgroup is itself and it has non-abelian composition factors. Consequently, the symmetric group S5 is also nonsolvable. More generally, the alternating groups An are nonsolvable for n ≥ 5, while An for n ≤ 4 are solvable. Similarly, Sn is solvable for n ≤ 4 and nonsolvable for n ≥ 5.

Connections and significance: Nonsolvability is a central concept in Galois theory, where the solvability of polynomials by radicals corresponds to the solvability of their Galois groups. The general quintic is not solvable by radicals because its typical Galois group is S5, which is not solvable.