Center is quasiautomorphism-invariant

Statement
The center of a group is a quasiautomorphism-invariant subgroup: it is invariant under all quasiautomorphisms of the group.

Center
Let $$G$$ be a group. The center of $$G$$, denoted $$Z(G)$$, is defined as follows:

$$Z(G) := \{ g \in G \mid gh = hg \ \forall \ h \in G \}$$.

In other words, $$Z(G)$$ is the set of those elements of $$G$$ that commute with every element of $$G$$.

Quasiautomorphism
Let $$G$$ and $$H$$ be groups. A function $$\varphi:G \to H$$ is termed a quasihomomorphism of groups if whenever $$a,b \in G$$ commute, we have $$\varphi(ab) = \varphi(a)\varphi(b)$$.

A function from a group to itself is termed a quasiautomorphism if it is a quasihomomorphism and has a two-sided inverse that is also a quasihomomorphism.

Quasiautomorphism-invariant subgroup
A subgroup of a group is termed quasiautomorphism-invariant if for every quasiautomorphism of the group, the subgroup gets mapped to within itself.

Related subgroup properties satisfied by the center

 * Center is characteristic
 * Center is strictly characteristic
 * Center is direct projection-invariant

Related subgroup properties not satisfied by the center

 * Center not is I-characteristic
 * Center not is fully characteristic