The Division Operation and The Extended Alpha Group: Exploring Deeper Dimensions

In the historical development of various fields of mathematics, significant advances have occurred in areas such as algebra, abstract algebra, group theory, and numerous other mathematical and scientific domains. Contributions from mathematicians such as Diophantus, Goldbach, Euler, Girolamo Cardano, Johannes Kepler, Poncelet, Henri Poincaré, George Cantor, Felix Klein, David Hilbert, and Hermann Weyl have been fundamental, particularly in the pursuit of increasingly complex and deeper structures within geometry and topology. In this work, the division operation in the Alpha group is defined by analogy with the Kronecker tensor product. The representation of quaternion theory, based on De Moivre’s theorem, is employed for the construction of the matrices. The Alpha Group division operation is then applied to analyze the various tensor metrics resulting from plane rotations over the interval from $0$ to $2\pi$ radians. Since the general transformation kernel of the $4 \times 4$ matrix is defined within the Alpha group, it is possible to observe the variability associated with the tangent and cotangent functions that constitute the transformation matrix. The Alpha group, defined through a generalized division operation, thus provides a geometric and topological representation of infinity via the kernel transformation of the $4 \times 4$ matrix. Ultimately, this work seeks to connect the ideas developed by Poncelet and Cantor regarding the formation of imaginary elements in infinite projections with the concept of different types of infinity, as interpreted through the application of group theory.