NP u -digital groups on real quaternions
Dae-Woong Lee, H. S. Lee, Sunyoung Lee, Jeong-Eun Lim, Seonjae Woo · Communications in Algebra · 2025
In this study, we introduce NPu-digital groups based on the real quaternions by formulating them as pointed digital images in the set Z4 of all lattice points in the four-dimensional Euclidean space R4. 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 NPu-digital groups for u∈{1,2}. More specifically, we construct a group X of order 48 as a pointed digital image which is not an NPu-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 NPu-digital groups by explicitly listing and analyzing each of the thirty five representatives with respect to various adjacency relations κv on Z4 for u∈{1,2} and v∈{1,2,3,4}.