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Non-linear electrodynamics arising in the frames of field theories innon-commutative space-time is examined on the base of theRiemann-Silberstein-Majorana-Oppenheimer formalism. The problem ofform-invariance of the non-linear constitutive relations governed by sixnon-commutative parameters \theta_{kl} \sim {\bf K} = {\bf n} + i {\bf m} isexplored in detail on the base of the complex orthogonal group theory SO(3.C).Two Abelian 2-parametric small groups, isomorphic to each other in abstractsense, and leaving unchangeable the extended constitutive relations atarbitrary six parameters \theta_{kl} of effective media have been found, theirrealization depends explicitly on invariant length {\bf K}^{2}. In the case ofnon-vanishing length a special reference frame in which the small group has thestructure SO(2) \otimes SO(1,1) has been found. In isotropic case no suchreference frame exists. The way to interpret both Abelian small groups inphysical terms consists in factorizing corresponding Lorentz transformationsinto Euclidean rotations and boosts. In the context of general study of variousdual symmetries in non-commutative field theory, it is demonstrated explicitlythat the non-linear constitutive equations in non-commutative electrodynamicsare not invariant under continuous dual rotations, instead only invarianceunder discrete dual transformation exists.

The Lorentz Group, Noncommutative Space-Time, and Nonlinear Electrodynamics in Majorana-Oppenheimer Formalism