Composition Series; Solvable and Nilpotent Groups

A finite group is built from simple pieces the way an integer is built from primes. The composition series makes this precise, the Jordan–Hölder theorem guarantees the pieces are well-defined, and the special classes of solvable and nilpotent groups measure how far a group is from being simple. Solvability is exactly the property that governs which polynomial equations are solvable by radicals. This page builds on sylow.md, permutation-groups.md, and the ideal-theoretic parallel in lie-algebras.md.

Normal and composition series

A subnormal series is a chain $$\{e\}=G_0◃G_1◃\cdots ◃G_k=G,$$ each term normal in the next. The quotients $G_{i+1}/G_i$ are the factors. It is a composition series if every factor is simple (no further refinement is possible). Every finite group has one.

Jordan–Hölder theorem. Any two composition series of a finite group have the same length and the same multiset of composition factors, up to reordering and isomorphism.

So the simple composition factors are an invariant of $G$ — its "prime factorisation". This is precisely why classifying the finite simple groups (simple-groups.md) classifies the building blocks of all finite groups (the assembly problem — how to combine them via extensions — is separate and harder).

Solvable groups

$G$ is solvable if it has a subnormal series with abelian factors — equivalently, if its composition factors are all cyclic of prime order. A cleaner test uses the derived series: with the commutator subgroup $G^{\prime }=[G,G]=⟨gh{g}^{-1}{h}^{-1}⟩$ (the smallest normal subgroup with abelian quotient), set $G^{(0)}=G$, $G^{(n+1)}=[G^{(n)},G^{(n)}]$. Then $$G\text{ is solvable}⟺G^{(n)}=\{e\}\text{ for some }n.$$

Facts:

• Subgroups, quotients, and extensions of solvable groups are solvable.
• Every abelian group is solvable; every $p$-group is solvable (below); every group of order $p^a{q}^b$ is solvable (Burnside, proved via characters).
• Feit–Thompson theorem: every group of odd order is solvable — a landmark 255-page proof.
• $S_n$ is solvable iff $n\le 4$. Since $A_5$ is simple non-abelian (permutation-groups.md), $S_5$ is unsolvable.

Galois connection. A polynomial is solvable by radicals iff its Galois group is solvable. The general quintic has Galois group $S_5$, which is unsolvable — hence the Abel–Ruffini theorem. This is the historical reason the word "solvable" was chosen, and the deepest application of finite group theory outside itself.

Nilpotent groups

$G$ is nilpotent if its lower central series $$G={\gamma }_1\supseteq {\gamma }_2\supseteq \cdots ,{\gamma }_{i+1}=[G,{\gamma }_i],$$ reaches $\{e\}$ in finitely many steps. Nilpotent is strictly stronger than solvable. Equivalent characterisations for finite $G$:

• $G$ is the direct product of its Sylow subgroups;
• every Sylow subgroup is normal ($n_p=1$ for all $p$, in the language of sylow.md);
• every maximal subgroup is normal; the ascending central series reaches $G$.

In particular every finite $p$-group is nilpotent — a consequence of the non-trivial-centre fact from the class equation. The hierarchy is $$\text{abelian}⊊\text{nilpotent}⊊\text{solvable}⊊\text{all finite groups}.$$ ($S_3$ is solvable but not nilpotent; $A_5$ is not even solvable.) The same three tiers reappear for Lie algebras — abelian ⊂ nilpotent ⊂ solvable, with the semisimple algebras as the opposite extreme — in lie-algebras.md.

References

• Dummit & Foote, Abstract Algebra, §3.4, §6.1.
• Rotman, An Introduction to the Theory of Groups, Ch. 5, 8.
• Isaacs, Finite Group Theory, Ch. 3–4.