# Transitively normal not implies central factor

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., transitively normal subgroup) neednotsatisfy the second subgroup property (i.e., central factor)

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## Statement

It is possible to have a group and a transitively normal subgroup of (i.e., any normal subgroup of is normal in ) that is not a central factor of .

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8, cyclic maximal subgroup of dihedral group:D8`

Consider the dihedral group of order eight:

.

Suppose is the cyclic subgroup of order four:

.

Then:

- 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 .