Solvable group

A group is solvable if repeated commutator subgroups eventually become trivial. Equivalently, it has a finite normal series whose successive quotients are abelian.

Derived subgroup: The subgroup generated by all commutators of a group. Iterating this subgroup operation defines the derived series used to test solvability.
Commutator: An element measuring how far two group elements are from commuting. Commutators generate each step of the derived series.
Galois theory: The study of field extensions through groups of field automorphisms. It translates solvability of a polynomial's Galois group into solvability by radicals.
Simple group: A nontrivial group with no proper nontrivial normal subgroups. Nonabelian simple groups are the basic obstructions to finite-group solvability.
Commutator subgroup: The subgroup generated by elements measuring the failure of pairs of group elements to commute. It is the first derived subgroup and captures the group's nonabelian part.
Derived length: The number of nontrivial derived-series steps before a group becomes trivial. It measures how many rounds of commutators solvability requires.
Solvability by radicals: The expression of polynomial roots using arithmetic operations and nested radicals. A polynomial over a suitable field is solvable by radicals precisely when its Galois group is solvable.
Perfect group: A group equal to its own commutator subgroup. Its derived series cannot decrease, so a nontrivial perfect group is not solvable.
Derived series: A descending sequence formed by repeatedly taking the derived subgroup. Solvability means this sequence reaches the trivial subgroup after finitely many steps.
Abelianization: The largest abelian quotient of a group, obtained by dividing out its commutator subgroup. Each derived-series step passes to the abelianization of the preceding subgroup.

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