Transitively normal not implies central factor
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., transitively normal subgroup) need not satisfy the second subgroup property (i.e., central factor)
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Example of the dihedral group
Consider the dihedral group of order eight:
Suppose is the cyclic subgroup of order four:
- is a transitively normal subgroup of : is normal (it has index two in ), and its proper subgroups (the trivial subgroup, and the subgroup ) are all normal in as well.
- is not a central factor of : Conjugation by gives an automorphism of that sends to , and is hence not an inner automorphism of .