Hypercomplex Group Theory

Abstract

Due to the noncommutative nature of quaternions and octonions we introduce barred operators. This objects give the opportunity to manipulate appropriately the hypercomplex fields. The standard problems arising in the definitions of transpose, determinant and trace for quaternionic and octonionic matrices are immediately overcome. We also investigate the possibility to formulate a new approach to Hypercomplex Group Theory (HGT). From a mathematical viewpoint, our aim is to highlight the possibility of looking at new hypercomplex groups by the use of barred operators as fundamental step toward a clear and complete discussion of HGT.