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 RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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).

Relation with other properties

Stronger properties

• Cyclic group

Weaker properties

• Metacyclic group
• A-group

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
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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.