Magnetic translation groups as group extension

TL;DR

This paper classifies magnetic translation groups as group extensions of cyclic groups by Abelian groups, using the Mac Lane method, with examples involving U(1) and Z_n.

Contribution

It systematically determines all possible extensions of cyclic groups by Abelian groups, identifying those that correspond to magnetic translation groups.

Findings

All factor systems for the extensions are classified.

Examples with G=U(1) and G=Z_n are explicitly discussed.

Magnetic translation groups are characterized as specific group extensions.

Abstract

Extensions of a direct product T of two cyclic groups Z_n1 and Z_n2 by an Abelian (gauge) group G with the trivial action of T on G are considered. All possible (nonequivalent) factor systems are determined using the Mac Lane method. Some of resulting groups describe magnetic translation groups. As examples extensions with G=U(1) and G=Z_n are considered and discussed.