Group in which every fully invariant subgroup is verbal
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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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 | |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 | |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 |