Nonsolvability

Nonsolvability is a property of a group that is not solvable. A group G is called solvable if it has a finite chain of subgroups G = G0 ⊇ G1 ⊇ ... ⊇ Gn = {e} where each Gi+1 is normal in Gi and the quotient Gi / Gi+1 is abelian. Equivalently, the derived series G^(0) = G, G^(n+1) = [G^(n), G^(n)], eventually reaches the trivial group; equivalently, a finite group is solvable if its composition factors are cyclic of prime order.

Nonsolvability arises when no such abelian-series exists. It is a central notion in the study of group structure and has important consequences in areas such as Galois theory, where the solvability of a polynomial by radicals is linked to the solvability of its Galois group.

A classic example of a nonsolvable finite group is the symmetric group on five elements, S5. Its derived subgroup is A5, and the derived subgroup of A5 is A5 itself, so the derived series never reaches the identity. Consequently S5 is not solvable, and A5 is the smallest non-solvable simple group. Many other non-abelian simple groups are nonsolvable, and in general, the presence of non-abelian simple quotients in a group's structure prevents solvability.

For finite groups, solvability corresponds to having a well-behaved, decomposable composition structure; for infinite groups, the concept extends similarly, with many important nonsolvable examples among infinite groups as well.