# Difference between revisions of "Group in which every maximal subgroup is normal"

From Groupprops

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| [[Weaker than::nilpotent group]] || has a [[central series]] of finite length || [[nilpotent implies every maximal subgroup is normal]] || [[every maximal subgroup is normal not implies nilpotent]] || {{intermediate notions short|group in which every maximal subgroup is normal|nilpotent group}} | | [[Weaker than::nilpotent group]] || has a [[central series]] of finite length || [[nilpotent implies every maximal subgroup is normal]] || [[every maximal subgroup is normal not implies nilpotent]] || {{intermediate notions short|group in which every maximal subgroup is normal|nilpotent group}} | ||

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+ | | [[Weaker than::group in which every subgroup is subnormal]] || every subgroup is a [[subnormal subgroup]] || [[every subgroup is subnormal implies every maximal subgroup is normal]] || || {{intermediate notions short|group in which every maximal subgroup is normal|group in which every subgroup is subnormal}} | ||

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| [[Weaker than::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]] || {{intermediate notions short|group in which every maximal subgroup is normal|hypercentral group}} | | [[Weaker than::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]] || {{intermediate notions short|group in which every maximal subgroup is normal|hypercentral group}} |

## Latest revision as of 00:56, 26 December 2015

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