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  "body": "\n## Statement\n\nLet $G$ be an [[abelian group|082GHNPJ53]].\nThen $G$ is [[solvable|0FXJ7YFMMX]].\n\n## Remarks\n\nThe series $\\{e\\} \\trianglelefteq G$ has one factor, $G$ itself, and this factor is abelian.\nEquivalently, the commutator subgroup $G^{(1)}$ is already $\\{e\\}$.\nThe converse is false: $S_3$ is solvable, by the series $\\{e\\} \\trianglelefteq A_3 \\trianglelefteq S_3$, but it is not abelian.\n\n## Explanations\n\n### L1 human\n\nAn abelian group is already one abelian layer.\nSo it is solvable with a series of length one.\nSolvable groups generalise abelian groups.\n",
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