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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]].\nThe group $G$ is a *Z-group* when, for every prime $p$, every [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ is [[cyclic|0KVGH0J5B3]].\n\n## Remarks\n\nSince the Sylow $p$-subgroups of $G$ are conjugate, it suffices to check one of them for each prime $p$ dividing the order of $G$.\nExample: a group of square-free order is a Z-group, because each of its Sylow subgroups has prime order and so is cyclic.\nIn particular $S_3$, of order $6$, is a Z-group.\nNon-example: the [[quaternion group|0NRJGW864D]] $Q_8$ is not a Z-group, because it is its own Sylow $2$-subgroup and it is not cyclic.\nEvery Z-group is [[solvable|0FXJ7YFMMX]], and every subgroup and every quotient of a Z-group is again a Z-group.\n\n## Explanations\n\n### L1 human\n\nA finite group is a Z-group when all its Sylow subgroups are cyclic.\n",
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