Nilpotent Covers of Dihedral Groups

Let G be a group, and let c ∈ Z + ∪ ∞. We let σ c (G) be the maximal size of a subset X of G such that, for any distinct x 1, x 2 ∈ X, the group ⟨ x 1, x 2 ⟩ is not c -nilpotent; similarly we let Σ c (G) be the smallest number of c -nilpotent subgroups of G whose union is equal to G. In this note we study D 2 k, the dihedral group of order 2 k. We calculate σ c (D 2 k) and Σ c (D 2 k), and we show that these two numbers coincide for any given c and k.