Characteristic not implies strictly characteristic
From Groupprops
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., strictly characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about characteristic subgroup|Get more facts about strictly characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not strictly characteristic subgroup|View examples of subgroups satisfying property characteristic subgroup and strictly characteristic subgroup
Contents
Statement
A characteristic subgroup of a group need not be a strictly characteristic subgroup.
Partial truth
It is true that for any finite group and more generally for any Hopfian group, characteristic subgroups are always strictly characteristic. This is because for Hopfian groups, surjective endomorphisms are the same thing as automorphisms. For full proof, refer: Characteristic equals strictly characteristic in Hopfian
Proof
Example constructed by Baer
This example is somewhat complex.
References
Journal references
- The higher commutator subgroups of a group by Reinhold Baer, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Page 143 - 160(Year 1944): This paper compares invariance properties such as normal subgroup, characteristic subgroup, strictly characteristic subgroup, and fully invariant subgroup.^{Full text (PDF)}
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