# Z-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## Definition

### Symbol-free definition

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

### 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 Finite group with periodic cohomology|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
View a complete list of subgroup-closed group properties

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.