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
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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
group in which every subgroup is subnormal	every subgroup is a subnormal subgroup	every subgroup is subnormal implies every maximal subgroup is normal		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