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A complete account is given of a question which is of great interest to specialists in mathematical logic and number theory.

CONTENTSIntroductionChapter I. Formal groups and Dieudonne modules; basic concepts1. Groups in categories2. Algebraic and formal groups. Bialgebras3. The structure of commutative artinian groups4. The Dieudonne module of a formal group5. CommentsChapter II. Dieudonne modules; classification up to isogeny1. Reduction of the problem2. Modules over the ring A3. A technical result4. Classification of formal groups up to isogeny5. CommentsChapter III. Dieudonne modules; classification up to isomorphism1. Statement of the problem2. Auxiliary results3. The algebraic structure on the module space4. The structure of isosimple modules; subsidiary reduction5. The structure of isosimple modules; proof of the first finiteness theorem6. The second finiteness theorem7. Cyclic isosimple modules; the component of maximal dimension8. Classification of two-dimensional modules9. CommentsChapter IV. Algebroid formal groups and abelian varieties1. General results2. The formal structure of abelian varieties; preliminary reduction3. The formal structure of abelian varieties; the fundamental theorem4. Weakly algebroid groups5. Remarks and examples6. CommentsReferences

A quantum mechanical method is used to calculate the probability of electron transfer via a bridge species. The calculation shows that the Arrhenius energy exponent is determined by the highest intersection point of the potential energy surface of the intermediate state either with that of the initial or with that of the final state. For suitable relative positions of the various terms, this energy is lower than the energy of the intersection point between potential energy surfaces of the initial and final states, corresponding to a direct electron transfer reaction. Thus the electron transfer processes through bridge systems can be more effective than direct electron transfer.