# Z-group

From Groupprops

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 propertiesVIEW 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:

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

- Every Sylow subgroup of is cyclic
- There exist cyclic subgroups and of such that is normal, and are permutable complements, and their orders are relatively prime (here is the normal Hall subgroup and 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.