Marginal homomorphism

Definition
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 $$G_2$$.

Particular cases

 * 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.

Related notions

 * Category of groups with marginal homomorphisms