DIFFERENCE BASES IN DIHEDRAL GROUPS

A subset b of a group g is called a {em‎ ‎difference basis} of g if each element g∈g can be written as the‎ ‎difference g=ab^−1 of some elements a,b∈b‎. ‎the smallest‎ ‎cardinality |b| of a difference basis b⊂g is called the {em‎ ‎difference size} of g and is denoted by δ[g]‎. ‎the fraction ‎‎‎ð[g]:=δ[g]/√|g| is called the {em difference characteristic} of g‎. ‎we prove that for every n∈n the dihedral group‎ ‎d2n of order 2n has the difference characteristic‎ ‎√2 ≤ð[d2n]≤48/√586 ≈1.983‎. ‎moreover‎, ‎if n≥2⋅10^15‎, ‎then ð[d2n]<4/√6 ≈1.633‎. ‎also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤80‎.