Center is strictly characteristic

Statement
The center of a group is always a strictly characteristic subgroup: any surjective endomorphism of the whole group sends the center to within itself.

Weaker facts: weaker subgroup properties satisfied
(Logically, the center satisfies all weaker properties, but we single out some important ones here).

Proof
Given: A group $$G$$, with center $$Z = Z(G)$$. A surjective endomorphism $$\sigma:G \to G$$.

To prove: $$\sigma(Z) \le Z$$.

Proof: Suppose $$g \in \sigma(Z)$$. We need to show that for any $$h \in G$$, $$gh = hg$$.

Since $$g \in \sigma(Z)$$, there exists $$a \in Z$$ such that $$g = \sigma(a)$$. Further, since $$\sigma$$ is surjective, there exists $$b \in G$$ such that $$\sigma(b) = h$$. Since $$a \in Z$$, we have:

$$ab = ba$$.

Applying $$\sigma$$ to both sides and using the property that $$\sigma$$ is a homomorphism yields:

$$\sigma(a)\sigma(b) = \sigma(b)\sigma(a) \implies gh = hg$$

as required.