Central factor implies normal

Verbal statement
Any central factor of a group is a normal subgroup.

Property-theoretic statement
The subgroup property of being a central factor is stronger than the subgroup property of being a normal subgroup.

Normal subgroup

 * Definition in terms of function restriction expressions
 * Definition in terms of normalizer

Central factor

 * Definition in terms of function restriction expression
 * Definition in terms of centralizer

Proof
The function restriction expression for normality is:

Inner automorphism $$\to$$ Function

The function restriction expression for central factor is:

Inner automorphism $$\to$$ Inner automorphism

Since the left sides in both cases are the same, and the right side for central factor is stronger than the right side for normality, the property of being a central factor is stronger than the property of being a normal subgroup.