On the structure of certain $\Gamma$-difference modules

This is a largely expository paper, providing self-contained account on the results of Schäfke and Singer (2017, 2019), in cases denoted there 2Q 2M. These papers supplied new proofs to main theorems Adamczewski Bell (2017) Bézivin Boutabaa (1992), rationality power series satisfying pair independent $q$-difference, or Mahler, equations. We emphasize language $\Gamma$ -difference modules, instead difference equations systems. Although two mentioned above this only semantic change, we also treat case, which may be labeled 1M1Q. Here group generalized dihedral rather than abelian, inadequate. In last section explain how generalize case finite characteristic.