  In this paper we study the description of the functional graphs associated
with the power maps over finite groups. We present a structural result which
describes the isomorphism class of these graphs for abelian groups and also for
flower groups, which is a special class of non abelian groups introduced in
this paper. Unlike the abelian case where all the trees associated with
periodic points are isomorphic, in the case of flower groups we prove that
several different classes of trees can occur. The class of central trees (i.e.
associated with periodic points that are in the center of the group) are in
general non-elementary and a recursive description is given in this work.
Flower groups include many non abelian groups such as dihedral and generalized
quaternion groups, and the projective general linear group of order two over a
finite field. In particular, we provide improvements on past works regarding
the description of the dynamics of the power map over these groups.
