# 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

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

## 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) |