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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe group $G$ is *supersolvable* when it has a chain of subgroups $\\{e\\} = G_0 \\le G_1 \\le \\cdots \\le G_n = G$ with this property.\nEach $G_i$ is a [[normal subgroup|0CB0DX5BVV]] of $G$ and each [[quotient group|0KYX2N69J0]] $G_i/G_{i-1}$ is [[cyclic|0KVGH0J5B3]].\n\n## Remarks\n\nExample: $S_3$ is supersolvable with the series $\\{e\\} \\le A_3 \\le S_3$.\nHere $A_3$ and $S_3$ are normal in $S_3$, $A_3 \\cong \\Z/3\\Z$ and $S_3/A_3 \\cong \\Z/2\\Z$.\nNon-example: $A_4$ is not supersolvable.\nIts only normal subgroups are $\\{e\\}$, the Klein four-group $V$ and $A_4$.\nSo the first term of such a series that differs from $\\{e\\}$ is $V$ or $A_4$, and the factor at that step is $V$ or $A_4$, which is not cyclic.\nEvery finitely generated [[nilpotent group|0Q7MMQHC1P]] is supersolvable, and every supersolvable group is [[polycyclic|9RBA56NXX3]], hence [[solvable|0FXJ7YFMMX]].\nSome authors write supersoluble.\n\n## Explanations\n\n### L1 human\n\nA group is supersolvable when it can be built from cyclic layers, with every partial stack normal in the whole group.\n",
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