Group in which every maximal subgroup is normal
From Groupprops
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
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 |
