The efficiency on 2-generators of semi-direct product of groups

Abstract

Let G be a semi-direct product of K by A where K and A are both cyclic groups of order n(nεN) and p (p is a prime), respectively. Then we prove that G has an efficient presentation on the minimal number, that is 2, of generators. After all, as an application of our main result, we give the efficiency of the dihedral group Dm and the metacylic group of order 2m ( m≥ 4 and m is even).