# Group in which every characteristic subgroup is strictly characteristic

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 characteristic subgroup is strictly characteristic is a group with the property that every characteristic subgroup of the group is a strictly characteristic subgroup.

Note that since characteristic not implies strictly characteristic, not all groups have this property.

## 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 (characteristic subgroup) satisfies the second property (strictly characteristic 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 impliation failure) Intermediate notions
Hopfian group every surjective endomorphism is an automorphism |FULL LIST, MORE INFO
Group in which every characteristic subgroup is fully invariant every characteristic subgroup is a fully invariant subgroup |FULL LIST, MORE INFO
Characteristically simple group nontrivial; no proper nontrivial characteristic subgroup |FULL LIST, MORE INFO