On complete decompositions of dihedral groups

Abstract

Let G be a finite non-abelian group and B1, …, Bt be nonempty subsets of G for integer t ≥ 2. Suppose that B1, …, Bt are pairwise disjoint, then (B1, …, Bt) is called a complete decomposition of G of order t if the subset product Bi1 … Bit = {bi1 … bit | bij ∈ Bij, j = 1,2, …,t} coincides with G, where {Bi1 … Bit} = {B1, …, Bt} and the Bij are all distinct. Let D2n = ‹r,s| rn = s2 = 1, rs =srn-1› be the dihedral group of order 2n for integer n ≥3. In this paper, we shall give the constructions of the complete decompositions of D2n of order t, where 2 ≤ t ≤ n.