Difference bases in dihedral groups

Abstract. A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as the difference g = ab -1 of some lements a; b ∈ B. The smallest cardinality ∣B∣ of a difference basis B ⊂ G is called the difference size of G and is denoted by Δ[G]. The fraction δ[G] := Δ[G]=√ jGj is called the difference characteristic of G. We prove that for every n 2 N the dihedral group D2n of order 2n has the difference characteristic p2 g[D2n] p48586 ≉ 1:983. Moreover, if n ≥ 2 10^15^, then δ[D2n] < p4 6 1:633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤ 80.