Solvable Group

A solvable group (or soluble group) is a group that can be constructed from abelian groups through a finite sequence of extensions. Solvability is the group-theoretic formalization of the intuition that a structure can be decomposed into simpler, commutative pieces.

The Feit-Thompson theorem states that every finite group of odd order is solvable. This result was foundational for the classification of finite simple groups, as it eliminated a vast class of potential simple groups from consideration. The concept originated in the work of Évariste Galois, who proved that a polynomial equation is solvable by radicals if and only if its Galois group is solvable.

The name 'solvable' is not metaphorical. It is literal. A group is solvable precisely when the equations it encodes are solvable in the algebraic sense. This is not a coincidence of terminology; it is the historical root of the entire concept. Galois did not invent group theory to study symmetry. He invented it to study which equations could be solved.