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  "body": "\n## Definition\n\nA [[group|G5S1M9]] $G$ is solvable when it has a [[subnormal series|0ZZR8D7Y1Z]] $\\{e\\} = G_0 \\trianglelefteq G_1 \\trianglelefteq \\cdots \\trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is [[abelian|082GHNPJ53]].\n\n## Remarks\n\nEquivalently, the [[derived series|0Q5G5DJ4P2]] of $G$ reaches $\\{e\\}$ after finitely many steps.\nA finite group is solvable if and only if its [[composition factors|0PHTC93M58]] are all cyclic of prime order.\nThe group $S_4$ is solvable, by the series $\\{e\\} \\trianglelefteq V \\trianglelefteq A_4 \\trianglelefteq S_4$.\nThe name comes from Galois theory: a polynomial equation over $\\mathbb{Q}$ is solvable by radicals if and only if its Galois group is solvable.\n\n## Explanations\n\n### L1 human\n\nA solvable group can be built from abelian groups in finitely many layers.\nAbelian groups are easy to understand, so solvable groups are close to easy.\nThe smallest group that is not solvable is $A_5$, of order $60$.\n",
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