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13 P R O P O S I T I O N . Let Q be a g r o u p a n d E v e r y c e n t r a l e x t e n s i o n c l a s s of a stem e x t e n s i o n PROOF. (e I) = ~ . hence . el) = O The n a t u r a l l t y of e I - ~(e) the i d e n t i t y of U in G extension f a c t o r s as an e p l m o r p h l s m and be any e x t e n s i o n exists by T h e o r e m of = 0 . 8 a n d is a e. , w h e n a p p l i e d to , the map W e now invoke 8. (e I) Q is i n d u c e d e ~ Ext(Qab,N) of e x t e n s i o n s , Since by where e I - V(e) eI , we find is stem.

E and ~, there is a unique M(Q) e ~ M(Q)~ . It will be a great advantage 29 that we may choose given problem. g. the standard the r e l a t i o n s h i p between cohomology treatment M(Q) Alternatively, group. 1LEMMA. 1) There of abelian the abelian (~,B,~): eI " e2 subquotients ally, PROOF. 10. anyway, The with the ] e as in of induced G i , for than by B on the i=1,2 . Actu- ~. As for the last assertion, morphism is an of extensions n(g) ~ ~2N2 ~2N2/[~2N2,G2 ] [~g1,~g2] of extensions and to a m o r p h i s m only on ?

S,F] for each fixed map surjective. )e,,l~,, that r~a-ICn). of d i a g r a m s • N . e. F/IS,F] is given by , whence w h i c h is n a t u r a l w i t h r e s p e c t groups, c([F,F] ? [S,F] e - e c(f,r) in the first v a r i a b l e Define If c Gab @ N - [R,F]/[S,F] ~(g[C,a] by Moreover × R/S e . 4) for the d e g e n e r a t e extension is an epimorphism. If A = A - 0 , we conM(A) is e v a l u a t e d at 43 the free p r e s e n t a t i o n for f,g ~ F . 6 E X A M P L E S . compute ~A A but Results: (i) T = M ( Y / m x Z/n) In the g r o u p determinant @ a I a ~ A~ Note: W e w r i t e multiplicatively.