---
tag: GJBAX2R2PG
kind: definition
name: Z-group
msc: [20D20, 20D10]
defines: 9VYFXJQ3DE
uses: [0053CQE9Y2, 0KVGH0J5B3]
sources:
  - url: https://en.wikipedia.org/wiki/Z-group
---

## Definition

Let $G$ be a finite [[group|G5S1M9]].
The group $G$ is a *Z-group* when, for every prime $p$, every [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ is [[cyclic|0KVGH0J5B3]].

## Remarks

Since the Sylow $p$-subgroups of $G$ are conjugate, it suffices to check one of them for each prime $p$ dividing the order of $G$.
Example: a group of square-free order is a Z-group, because each of its Sylow subgroups has prime order and so is cyclic.
In particular $S_3$, of order $6$, is a Z-group.
Non-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.
Every Z-group is [[solvable|0FXJ7YFMMX]], and every subgroup and every quotient of a Z-group is again a Z-group.

## Explanations

### L1 human

A finite group is a Z-group when all its Sylow subgroups are cyclic.