Marginal homomorphism

From Groupprops
Jump to: navigation, search

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