What is the difference between solvable and nilpotent groups?

Nilpotent groups have a central series and form a strictly smaller class; every nilpotent group is solvable but not conversely. Finite nilpotent groups are exactly direct products of their Sylow subgroups.

Why does the symmetric group on five letters fail to be solvable?

Its derived series stabilizes at the nontrivial alternating group on five letters, which is simple and non-abelian, so the series never reaches the trivial subgroup. This non-solvability is why the general quintic has no radical formula.

Solvable Group

A solvable group is one that can be built from abelian pieces through a chain of normal subgroups, a structural property that governs whether polynomial equations are solvable by radicals.

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Definition

A group is solvable if it has a finite subnormal series whose successive quotient groups are all abelian, equivalently if its derived series terminates at the trivial subgroup.

Scope

This topic covers the derived series and commutator subgroups, subnormal series with abelian factors, the equivalence of the various definitions of solvability, nilpotent groups as a stronger condition, and the role of solvable groups in Galois theory.

Core questions

What does it mean to build a group from abelian layers?
How do the derived series and subnormal series characterize solvability?
Which standard families of groups are solvable, and which are not?
Why is solvability the decisive condition for solving equations by radicals?

Key theories

Derived series characterization: A group is solvable exactly when its derived series, obtained by iterating the commutator subgroup, reaches the trivial group in finitely many steps.
Closure properties of solvable groups: Subgroups and quotient groups of solvable groups are solvable, and an extension of a solvable group by a solvable group is solvable, so solvability is preserved under the standard structural operations.
Solvability and radicals: A polynomial over a field of characteristic zero is solvable by radicals if and only if its Galois group is a solvable group, the criterion that proves the general quintic cannot be solved by radicals.

Clinical relevance

Solvable groups are the precise obstruction in the theory of equations: Galois's criterion connects solvability of a group to solvability of polynomials by radicals. The concept also organizes finite group theory, where the Feit-Thompson theorem shows every group of odd order is solvable.

History

The notion arose from Galois's study of which equations are solvable by radicals, where 'solvable' originally referred to the equation; the corresponding group-theoretic property kept the name. The Feit-Thompson theorem of 1963, that all groups of odd order are solvable, was a landmark in the classification of finite simple groups.

Key figures

Évariste Galois
Walter Feit
John G. Thompson

Seminal works

dummit2004
rotman1995
isaacs2008

Frequently asked questions

What is the difference between solvable and nilpotent groups?: Nilpotent groups have a central series and form a strictly smaller class; every nilpotent group is solvable but not conversely. Finite nilpotent groups are exactly direct products of their Sylow subgroups.
Why does the symmetric group on five letters fail to be solvable?: Its derived series stabilizes at the nontrivial alternating group on five letters, which is simple and non-abelian, so the series never reaches the trivial subgroup. This non-solvability is why the general quintic has no radical formula.