A Description of an Automorphism of a Non-Split Metacyclic p-Group

A map on a group is not necessarily an automorphism on the group. in this paper we study the necessary and sufficient conditions for a map on a non-split metacyclic p-group to be an automorphism, where we only consider p as an odd prime number. the metacyclic group can be defined by a presentation and it will be beneficial to have a direct relation between the parameters in the presentation and an automorphism of the group. we consider the action of an automorphism on the generators of the group mentioned. since any element of a metacyclic group will be mapped to an element of the group by an automorphism, we can conveniently represent the automorphism in a matrix notation. we then use the relations and the regularity of the non-split metacyclic p-group to find conditions on each entry of the matrix in terms of the parameters in its presentation so that such a matrix does indeed represent an automorphism