Group theory and generalizations Definition

Z-group

Definition

Let $G$ be a finite group. The group $G$ is a Z-group when, for every prime $p$ , every Sylow $p$ -subgroup of $G$ is cyclic.

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 $Q_8$ is not a Z-group, because it is its own Sylow $2$ -subgroup and it is not cyclic. Every Z-group is solvable, and every subgroup and every quotient of a Z-group is again a Z-group.

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Cyclic group concept
Sylow p-subgroup concept
Z-group definition

Tag	`GJBAX2R2PG`
URI	`m2://GJBAX2R2PG`
Kind	definition
Writing	W0 stub
MSC	20D20, 20D10
Defines	Z-group
Uses	Sylow p-subgroup, Cyclic group
Changed	2 November 2026

Z-group

Definition

Remarks

Reading path

Further reading

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