Digraphs From Power Mapping On Noncommutative Groups And Rings

A discrete dynamical system is consisting of a finite non-empty set X and a map from X into itself.The phase state or iteration graph of the dynamical system,denoted by G(X,f),is a digraph,whose vertex set is X and there is an arrow from a to b if and only if f(a)= b for a,b ? X.When X is a group or a ring and f maps x into its k-th power,we write G(X,k)instead of G(X,f)and call this digraph the power digraph.The research of iteration digraphs is involved with graph theory,ring theory and number theory.In this paper,we study the iteration graph over finite group and finite noncommutative ring.In Chapter I,we introduce the background of iteration and its application in other areas.In Chapter II,we first investigate the basic property of the iteration graph defined by a group endomorphism of a finite group H.These digraphs have a highly symmetric structure.Using the wreath product of groups,we determine the automorphism group of these digraphs.We also consider when the discrete dynamical system is a fixed point system.Through the Chinese Remainder Theorem and representing the endomorphism by an integer matrix,we give a necessary and sufficient condition for that G(H,f)is fixed point system.In Chapter III,we mainly study the k-th power iteration digraphs from the 2 x 2 matrix ring R over a finite field.We decomposite this digraph into two parts,one is consisting of singular matrices and the other is consisting of invertible matrices.We also give necessary and sufficient condition for that when G(R,k1)is isomorphic to G(R,k2).In the last section of this chapter,we study the power iteration digraph on semiproduct of cyclic group.