Automorphisms of Automorphism Group of Dihedral Groups

The automorphism group of a Dihedral group of order is isomorphic to the holomorph of a cyclic group of order . The holomorph of a cyclic group of order is a complete group when is odd. Hence automorphism groups of Dihedral groups of order are its own automorphism groups whenever is odd. In this paper, we prove that the result is also true for those Dihedral groups of order where is twice a prime number.