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This work treats a conjecture of Peter Schmid on the Tate cohomology of finite p-groups. We examine some cases in which the conjecture has an affirmative answer, as well as the relevance of this conjecture to studying automorphisms of finite p-groups.- The First chapter is an overview of the structure of p-groups.- The Scened chapter includes other types of p -groups that are related to the p-groups; which isthe following:- p-group semi-abelian and strongly semi-abelian.- Regular p-groups.- p-central p-groups.- Powerful p-groups.and we talked about power structure of p-groups.- In the Third chapter, which is the most important part of our work, which in turn discussesSchmid's conjecture by using cohomological technique, and some results about them. wherewe initially explained the cohomology of finite p-groups and As we have also mentioned forGaschütz and Uchida theorem on the triviality of the cohomology of finite p-groups.- In the last chapter we devoted the study on the existence of non-inner automorphism of p-power order in the finite p-groups and we find there is a relation between cohomologicallytrivial modules and the existence of non inner automorphism.