Digital topological groups

In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate categories, based on the details of the continuity required of the group multiplication. We define ${\text{NP}}_1$- and ${\text{NP}}_2$-digital topological groups, and investigate their properties and algebraic structure. The ${\text{NP}}_2$ category is very restrictive, and we give a complete classification of ${\text{NP}}_2$-digital topological groups. We also give many examples of ${\text{NP}}_1$-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.