  We prove that the relative commutator with respect to a subvariety of a
variety of Omega-groups introduced by the first author can be described in
terms of categorical Galois theory. This extends the known correspondence
between the Froehlich-Lue and the Janelidze-Kelly notions of central extension.
As an example outside the context of Omega-groups we study the reflection of
the category of loops to the category of groups where we obtain an
interpretation of the associator as a relative commutator.
