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 $\varphi (V^{*}(G_{1}))\leq V^{*}(G_{2})$ 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