Group in which every fully invariant subgroup is verbal

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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 fully invariant subgroup is verbal is a group satsfying the property that every fully invariant subgroup of the group is a verbal subgroup.

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 (fully invariant subgroup) satisfies the second property (verbal 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
Free group Reduced free group|FULL LIST, MORE INFO
Reduced free group quotient of a free group by a verbal subgroup fully invariant implies verbal in reduced free group |FULL LIST, MORE INFO
Free abelian group Reduced free group|FULL LIST, MORE INFO
Cyclic group |FULL LIST, MORE INFO
Homocyclic group product of isomorphic cyclic groups |FULL LIST, MORE INFO
Simple group nontrivial, no proper nontrivial normal subgroup |FULL LIST, MORE INFO
Characteristically simple group nontrivial, no proper nontrivial characteristic subgroup |FULL LIST, MORE INFO