Normal 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., normal subgroup) need not satisfy the second subgroup property (i.e., normal-potentially characteristic subgroup)
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Statement with symbols
- Normal not implies normal-extensible automorphism-invariant
- Normal-potentially characteristic implies normal-extensible automorphism-invariant
The proof follows directly 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 normal in : This is obvious.