NP_u-digital groups on real quaternions

In this study, we introduce (Formula presented.) -digital groups based on the real quaternions by formulating them as pointed digital images in the set (Formula presented.) of all lattice points in the four-dimensional Euclidean space (Formula presented.). To facilitate the algebraic structure, we define a notion of digital multiplications on these pointed digital images, inspired by the multiplicative operation in the division ring of real quaternions, thereby enabling the construction of (Formula presented.) -digital groups for (Formula presented.). More specifically, we construct a group X of order 48 as a pointed digital image which is not an (Formula presented.) -digital group, and establish an isomorphism between the group X as a pointed digital image and the binary octahedral group G, which is a well-known non-abelian group of order 48 in classical algebra. We enumerate 35 specific subgroups of X, and then investigate these subgroups thoroughly to be qualified (or non-qualified) as (Formula presented.) -digital groups by explicitly listing and analyzing each of the thirty five representatives with respect to various adjacency relations (Formula presented.) on (Formula presented.) for (Formula presented.) and (Formula presented.).