Worked solution: Abel–Ruffini theorem and the Galois solvability criterion (1824)

Step 4 of 8: Solvable groups: chains that always end in abelian steps

In plain words

Think of taking apart a complicated machine into a stack of simpler and simpler gearboxes, where every single gearbox in the stack, taken on its own, behaves like a basic, commutative dial (turning it left then right gives the same result as right then left). A group earns the label "solvable" exactly when it can be taken apart into such a stack, however many pieces that takes.

G \text{ solvable} \iff \exists\; 1 = G_r \triangleleft G_{r-1} \triangleleft \cdots \triangleleft G_0 = G \text{ with each } G_i/G_{i+1} \text{ abelian}

Detailed analysis

A group $G$ is solvable if there is a chain of subgroups $1=G_r\triangleleft G_{r-1}\triangleleft\cdots\triangleleft G_0=G$ , each normal in the next one up, whose successive quotients $G_i/G_{i+1}$ are all abelian (commutative). This is exactly the shape the correspondence in Step 3 produces from a radical tower: the subgroup chain $G=H_0\ge H_1\ge\cdots\ge H_m=1$ built there has abelian (in fact cyclic) quotients $H_{i-1}/H_i$ , so $G$ is solvable whenever $f$ is solvable by radicals.

Terms in this step

Solvable group: A group that can be broken down by a chain of normal subgroups into steps with abelian quotients; the name comes directly from this theorem linking it to solving equations by radicals.
Quotient group: The group $G/N$ formed from a group $G$ and a normal subgroup $N$ by treating elements that differ by something in $N$ as the same.

Knowledge used in this step

Group theory