The Summary Of Propertive Of Automorphism With Order P~n-1of The Elementary Abelian Group With Order P~n

In group theory, in addition to cyclic group, a kind of simple group is the primary Abel p-group, whose structure is very clear. Research on this group of automorphisms is already complete, its automorphism group is also already known. But the group has an interesting automorphism of high order, its existence can be proved by the theory of finite fields. For an elementary Abelian group of pn order, its automorphism is the general linear group of n dimension, GL(n,p). The group must have a pn-1order automorphism. Although the existence of this kind of automorphisms has been proved, what is the form? How do they act on group elements? There is no general rule. This thesis reviews the knowledge, and attempts to give the automorphism of pn-1order in the form of a matrix and group elements mode of action through the finite field theory. As an example, we give the automorphisms of order80of34order elementary Abelian p-groups in the form of a matrix, and we also give the primitive element set forming by their actions on group elements. This article is divided into three sections.The first section introduces the historical background of automorphism group.The second section introduces general theory showing the existence of automor-phism of order pn-1of an elementary Abelian group of order pn, and matrix forms of such kinds of automorphism. Also the method to determine sets of imprimitive elements of action on the group by this kind of automorphism.The third section gives an example of theory in the second section, which is the case of an elementary Abelian group with order81.