Group whose center is comparable with all normal subgroups

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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
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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

A group whose center is comparable with all normal subgroups is defined as a group whose center is a subgroup comparable with all normal subgroups. In other words, every normal subgroup is either a central subgroup (i.e., it is contained in the center) or is a subgroup containing the center.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal-comparable group any two normal subgroups are comparable center is normal |FULL LIST, MORE INFO
Abelian group center equals whole group |FULL LIST, MORE INFO
Centerless group center is trivial |FULL LIST, MORE INFO
Group in which every proper normal subgroup is central |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Group having no proper cocentral subgroup (in the non-abelian case)