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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe group $G$ is *polycyclic* when it has a [[subnormal series|0ZZR8D7Y1Z]]\n$$\\{e\\} = G_0 \\trianglelefteq G_1 \\trianglelefteq \\cdots \\trianglelefteq G_n = G$$\nin which each factor $G_i/G_{i-1}$ is [[cyclic|0KVGH0J5B3]].\n\n## Remarks\n\nEquivalently, $G$ is [[solvable|0FXJ7YFMMX]] and every subgroup of $G$ is [[finitely generated|082DWWXMRJ]].\nThe number of infinite cyclic factors does not depend on the series; it is the Hirsch length of $G$.\nExample: $\\Z^2$ is polycyclic, with the series $0 \\le \\Z \\times 0 \\le \\Z^2$ whose factors are both isomorphic to $\\Z$; its Hirsch length is $2$.\nNon-example: the additive group $\\Q$ is abelian, hence solvable, but it is not polycyclic.\nA polycyclic group is generated by lifts of generators of its factors, so it is finitely generated, while $\\Q$ is not finitely generated.\nEvery finite solvable group is polycyclic, and every [[supersolvable group|AWWM9RWETX]] is polycyclic.\n\n## Explanations\n\n### L1 human\n\nA group is polycyclic when it can be built from finitely many cyclic layers, each normal in the next.\n",
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