A group $G$ is called solvable if it has a subnormal series with abelian factors; that is, there are groups

$$\{1\} = G_0 \le G_1 \le G_2 \le \dots \le G_n = G$$

such that $G_i$ is normal in $G_{i + 1}$ and the factor group $G_{i + 1} / G_i$ is abelian for each $i$.

Solvable groups arise naturally in Galois theory, as a polynomial equation is solvable by radicals if and only if its Galois group is solvable.

Source: Solvable group.

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