# Marginal homomorphism

Suppose $\mathcal{V}$ is a subvariety of the variety of groups. Suppose $G_1$ and $G_2$ are groups (not necessarily in $\mathcal{V}$, and in fact, definitely not in the interesting cases). Suppose $\varphi:G_1 \to G_2$ is a homomorphism of groups. We say that $\varphi$ is a $\mathcal{V}$-marginal homomorphism if $\varphi(V^*(G_1)) \le V^*(G_2)$ where $V^*(G_1)$ and $V^*(G_2)$ denote respectively the $\mathcal{V}$-marginal subgroups of $G_1$ and $G_2$.
• In case $\mathcal{V}$ is the subvariety comprising only the trivial group, all homomorphisms are marginal homomorphisms.
• In case $\mathcal{V}$ is the subvariety comprising all groups, all homomorphisms are marginal homomorphisms.
• In case $\mathcal{V}$ is the subvariety comprising abelian groups only, the marginal homomorphism are precisely the central homomorphisms.