Z-group

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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.