---
tag: 9VYFXJQ3DE
kind: concept
name: Z-group
msc: [20D20, 20D10]
sources:
  - url: https://en.wikipedia.org/wiki/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.

## Explanations

### L1 human

For each prime $p$, a finite group has Sylow $p$-subgroups, the largest subgroups whose order is a power of $p$.
In a Z-group, each of these is generated by a single element.
For 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.
So $S_3$ is a Z-group, while the Klein four-group is not, since it is its own Sylow $2$-subgroup and it is not cyclic.