  In this paper, we examine the relations of two closely related concepts, the
digital Lusternik-Schnirelmann category and the digital higher topological
complexity, with each other in digital images. For some certain digital images,
we introduce $\kappa-$topological groups in the digital topological manner for
having stronger ideas about the digital higher topological complexity. Our aim
is to improve the understanding of the digital higher topological complexity.
We present examples and counterexamples for $\kappa-$topological groups.
