Z-group

Summary

A Z-group is a finite group in which every Sylow subgroup is cyclic. These groups are well understood: each one is solvable, and in fact metacyclic, that is, an extension of a cyclic group by a cyclic group. Every group of square-free order is a Z-group. The name Z-group is also used for some unrelated classes of groups; this node is about the finite groups with cyclic Sylow subgroups.