# Group in which every maximal subgroup is normal

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 is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

## Definition

A **group in which every maximal subgroup is normal** is a group satisfying the following equivalent conditions:

- Any maximal subgroup (i.e. any proper subgroup not contained in any other proper subgroup) is a normal subgroup
- Any maximal subgroup is a maximal normal subgroup
- There does not exist any maximal subgroup that is contranormal
- There does not exist any maximal subgroup that is self-normalizing

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

nilpotent group | has a central series of finite length | nilpotent implies every maximal subgroup is normal | every maximal subgroup is normal not implies nilpotent | Group in which every subgroup is subnormal, Group satisfying normalizer condition|FULL LIST, MORE INFO |

hypercentral group | hypercenter of the group (the point at which the transfinite upper central series stabilizes) is the whole group | hypercentral implies every maximal subgroup is normal | every maximal subgroup is normal not implies hypercentral | |FULL LIST, MORE INFO |

group satisfying normalizer condition | there is no proper self-normalizing subgroup | normalizer condition implies every maximal subgroup is normal | every maximal subgroup is normal not implies normalizer condition | |FULL LIST, MORE INFO |