Some constructions of complete decomposition of dihedral group

Abstract

Abstract Let G be a finite non-abelian group. For integer k ≥ 2, we let A 1,…,Ak be non-empty subsets of G. If A 1,…, Ak are pairwise disjoint and if the subset product A i 1 … Aik = {a i 1 …aik |aij ∈ Aij , j = 1,…, k} coincides with G, where the Aij are all distinct and {A i 1 ,…, Aik } = {A 1,…, Ak } then (A 1,…, Ak ) is called a (k, |A 1|,…, |Ak |)-complete decomposition of G. For integer n ≥ 3, let D2n be the dihedral group of order n. Let A, B be the subset of D2n . In this paper, we show some constructions, namely (2, |A|, |B|)-complete decomposition of D2n where | B | ∈ { 2 , 4 , … 2 ⌊ n 3 ⌋ , n − 1 , n } and |A| = 2n − |B|.