Z-group

Symbol-free definition
A finite group is termed a Z-group if it satisfies the following equivalent conditions:


 * Every Sylow subgroup of it is cyclic
 * There exists a cyclic normal Hall subgroup with a cyclic quotient group (viz a cyclic complement)

Definition with symbols
A finite group $$G$$ is termed a Z-group if it satisfies the following equivalent conditions:


 * Every Sylow subgroup of $$G$$ is cyclic
 * There exist cyclic subgroups $$N$$ and $$H$$ of $$G$$ such that $$N$$ is normal, $$N$$ and $$H$$ are permutable complements, and their orders are relatively prime (here $$N$$ is the normal Hall subgroup and $$H$$ is the complement).

Metaproperties
This follows from the fact that Sylow subgroups of the subgroup sit inside Sylow subgroups of the whole group.

This follows from the fact that quotient maps take Sylow subgroups to Sylow subgroups.

This follows from the fact that Sylow subgroups in the direct product arise as direct products of Sylow subgroups in the direct factors.