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

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## Statement

A normal subgroup of a group need not be a central factor.

## Related facts

- Normal not implies direct factor
- Central factor not implies direct factor
- Abelian normal not implies central

## Proof

`Further information: dihedral group:D8, Klein four-subgroups of dihedral group:D8, cyclic maximal subgroup of dihedral group:D8`

Suppose is the dihedral group of order eight, is the cyclic subgroup of order four in , and and are the two Klein four-subgroups. Then, the three subgroups , , and are all normal in . However, none of them are central factors. In fact, they are all self-centralizing subgroups of .