Characteristic not implies isomorph-normal in finite group
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., isomorph-normal subgroup)
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Statement with symbols
Example of the dihedral group
Further information: dihedral group:D8
Let be the dihedral group of order eight, given by:
Let be the center of . is a subgroup of order two generated by .
- is characteristic.
- is not isomorph-normal: The subgroup of is isomorphic to , but is not normal in , because conjugation by sends it to the subgroup .