By Prof. Dr. Michel Lazard (auth.)

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Is s u f f i c i e n t : (O,~') , (0,~') , (~o,O) If namely, ~o = O, we replace -5i- 8. 1 Lemma. inteqer Jn_If are Let i> 2 Assume that f elements Proof. 5). 11) P is P. 2). 7). 8). allows a curvilinear (An_ l ) , and vanishing w(f'(x,y)) is c u r v i l i n e a r (A n ) ~ see defines K I . T h e n we l , then f' is c u r v i l i n e a r , = ~(x) to a s s u m e n-bud take coefficients , defined us In p r o v i n g over that the the m o r p h i s m in d e g r e e s >i n . 10). 7). deg. then lemma f : D(I) this w a y defines C i C n ( X i , Y i) can apply w + Zi~ I and (necessarily n two m o r p h i s m s .

20 of the g reduces a formal module out I ..... n such . 17) , the b a s i c V, this symmetric ring K may theorem polynomials. remain unde- fined. CHAPTER FORMAL GROUPS I. 2 admits ables on G corresponding tities G ways in the c a t e g o r y , alternative by giving, theory qroups, products. with some of group (resp. (resp. commutative G n ~ G, word-functions or c o m m u Anyhow, extra a struc- descriptions. of commutative for e a c h w o r d - f u n c t i o n word-morphism, relatinq to d e f i n e wit h finite a structure from group BUDS in c a t e q o r i e s in a c a t e q o r y We can define group) AND equivalent is an o b j e c t ture w h i c h II group in n theory) in the c a t e g o r y .

O v e r a basic ring K contains, a n o n - z e r o nilpotent e l e m e n t of finite a d d i t i v e order. In the e x c e p t i o n a l case, a 2 = O and pa = O there is some (p prime); then a ~ K, such that x + y + ax~ is a non- c o m m u t a t i v e group law. 1), see [14] and [6]. C H A P T E R III THE G E N E R A L E Q U I V A L ~ C E i. 12). 11) . ca! groups, ~K(G_m) , w i t h the same u n d e r l y i n g uniform space, n a m e l y the set of m o r p h i s m s ~(DK,DK) with its (simple or order) -58topology (once the topological matter which one - there space and a uniform space; remember K .