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 .