Potentially characteristic not implies normal-potentially characteristic
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., potentially characteristic subgroup) need not satisfy the second subgroup property (i.e., normal-potentially characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about potentially characteristic subgroup|Get more facts about normal-potentially characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property potentially characteristic subgroup but not normal-potentially characteristic subgroup|View examples of subgroups satisfying property potentially characteristic subgroup and normal-potentially characteristic subgroup
- Normal not implies normal-potentially characteristic
- Normal not implies characteristic-potentially characteristic
- Potentially characteristic not implies characteristic-potentially characteristic
- Potentially characteristic not implies normal-extensible automorphism-invariant, which in turn follows from Normal not implies normal-extensible automorphism-invariant in finite and finite normal implies potentially characteristic.
- Normal-potentially characteristic implies normal-extensible automorphism-invariant
The proof follows from facts (1) and (2).
Example of the dihedral group
Further information: dihedral group:D8
Let be the dihedral group of order eight, and be one of the Klein four-subgroups.
- is not a normal-potentially characteristic subgroup of : Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of can be extended to an automorphism of for any group containing as a normal subgroup. But since there is an automorphism of not sending to itself, cannot be characteristic in .
- is potentially characteristic in : for instance, we can realize as the -Sylow subgroup of the symmetric group of degree four, in such a way that becomes characteristic.