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 $n (nvarepsilon N)$ 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 $D_m$ and the metacylic group of order 2m ( m$geq$ 4 and m is even).