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  "body": "\n## Summary\n\nA Z-group is a finite group in which every Sylow subgroup is cyclic.\nThese groups are well understood: each one is solvable, and in fact metacyclic, that is, an extension of a cyclic group by a cyclic group.\nEvery group of square-free order is a Z-group.\nThe name Z-group is also used for some unrelated classes of groups; this node is about the finite groups with cyclic Sylow subgroups.\n\n## Explanations\n\n### L1 human\n\nFor each prime $p$, a finite group has Sylow $p$-subgroups, the largest subgroups whose order is a power of $p$.\nIn a Z-group, each of these is generated by a single element.\nFor example, the symmetric group $S_3$ has order $6 = 2 \\cdot 3$, its Sylow subgroups have orders $2$ and $3$, and groups of prime order are cyclic.\nSo $S_3$ is a Z-group, while the Klein four-group is not, since it is its own Sylow $2$-subgroup and it is not cyclic.\n",
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