2-Generator golod p-groups by Timofeenko A.V.

By Timofeenko A.V.

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Prop. , det d2F(X,O) is a unit, and so d2F(X,0) is aa -27- invertlble matrix. /BX. = 0 for ~11 i,j. l O this implies f = O. , e~ = 8 o ~. g induction. , that g(X) is a homomorphism of formal groups. Now g(F(P)(xP,YP)) = gCFCX,Y) p) = fCFCX,Y)) = GCf(x),fCY)) = where F (p) is obtained from F by raising each coefficient to its pth p~er. }~ have then g(F(P)(x,Y)) = G(g(X),g(Y)). Since F(P)(x,Y) is a formal group (the map which sends each element of R into its pth p~er is an endomorphism of R), g is indeed a homomorphism of formal g%'OUpS.

E~ = 8 o ~. g induction. , that g(X) is a homomorphism of formal groups. Now g(F(P)(xP,YP)) = gCFCX,Y) p) = fCFCX,Y)) = GCf(x),fCY)) = where F (p) is obtained from F by raising each coefficient to its pth p~er. }~ have then g(F(P)(x,Y)) = G(g(X),g(Y)). Since F(P)(x,Y) is a formal group (the map which sends each element of R into its pth p~er is an endomorphism of R), g is indeed a homomorphism of formal g%'OUpS. If 8 = r o =(h), and D(~) # O, then h = ht(8) is called the height of 8 9 l~e define ht(O) = = .

Let {C be t h e c a t e g o r y whose o b j e c t s a r e t h e c o a l g e b r a s {Un' ~n' r ~n } and whose morphisms a r e t h e homomorphisms o f coalgebras. The l a s t p r o p o s i t i o n t h e n t e l l s j ~ § J ~ * y i e l d s an antlsomorphism ~ § ~ us t h a t t h e f ~ n c t o r of categories. Let F = F(X' ,X") be a power s e r i e s i n 2n i n d e t e r m i n a t e s x, - x"- v,, v. , w i t h zero c o n s t a n t t e r m . Let 8F E Hom~ ( R n , ~ n ) be t h e c o r r e s p o n d i n g homomorghism o f power series rings.

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