Group in which every permutable subgroup is normal

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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

Definition

A group in which every permutable subgroup is normal is a group such that every permutable subgroup of the group is a normal subgroup of the group.

Formalisms

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (permutable subgroup) satisfies the second property (normal subgroup), and vice versa.
View other group properties obtained in this way

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian group any two elements commute |FULL LIST, MORE INFO
Dedekind group every subgroup is a normal subgroup |FULL LIST, MORE INFO
Finite T-group |FULL LIST, MORE INFO