Z-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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View other semistandard definitions in group theory

Definition

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).

Equivalence of definitions

Further information: Every Sylow subgroup is cyclic implies metacyclic

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite cyclic group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
metacyclic group
A-group	all Sylow subgroups are abelian
finite group with periodic cohomology	all Sylow subgroups have rank one, i.e., any abelian p-subgroup is cyclic
Schur-trivial group	The Schur multiplier (which is the second homology group for trivial group action with integer coefficients) is trivial			|FULL LIST, MORE INFO

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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This follows from the fact that Sylow subgroups of the subgroup sit inside Sylow subgroups of the whole group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

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

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

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