Explore the concept of the one-dimensional random walk hypergroup in this 29-minute mathematics lecture. Delve into affine transformations and the ax+b group to understand symmetries of the one-dimensional integral lattice. Learn about representation theory by assigning 2x2 matrices to affine transformations of the line. Examine the space group of the one-dimensional integer lattice, involving semi-direct products and subgroup normalization. Investigate the random walk associated with this lattice as a graph and its corresponding hypergroup. Conclude with algebraic computations to determine expected positions after n steps in a nearest neighbor random walk. Gain insights into research-level mathematics through concrete calculations and examples.