Mac Lane method in the investigation of magnetic translation groups

Central extensions of the three-dimensional translation group T ∼- Z3 by the unitary group U(1) (a group of factors) are considered within the frame of the Mac Lane method. All nonzero vectors t gE T are considered to be generators of T. This choice leads to very illustrative relations between the Mac Lane method and Zakʹs approach to magnetic translation groups. It is shown that factor systems introduced by Zak and Brown can be realized only for the unitary group U(1) and for some of its finite subgroups.