Group in which any two normal subgroups are comparable

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Definition

A group is said to be a group in which any two normal subgroups are comparable if any two normal subgroups of the group can be compared, or in other words, if its lattice of normal subgroups is a totally ordered set. In other words, given any two normal subgroups of the group, one of them must lie completely inside the other.

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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Simple group no proper nontrivial normal subgroup
Cyclic group of prime power order cyclic group, order is a prime power

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which any two characteristic subgroups are comparable any two characteristic subgroups are comparable
group in which every normal subgroup is characteristic (at least for a finite group)
group whose center is comparable with all normal subgroups

Facts