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 NP1- and NP2-digital topological groups, and investigate their properties and algebraic structure. The NP2 category is very restrictive, and we give a complete classification of NP2-digital topological groups. We also give many examples of NP1-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.