By D. W. Anderson (auth.), Prof. Dr. Peter J. Hilton (eds.)

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**Example text**

The s i m p l e s t group e x t e n s i o n 0 -) T -~ K -~ W -~ 1 w i t h given T × W . w 2) = (tlt21~wlw2) wl t2 where extension map is the image K K-~ W t2 has a homomorphic natural product G and also Spin(n) splits. The converse does not split~ 2: It seems because years. approach ~ S0(n) Theorem subgroup For, Note of N does S0(n) Spin(n) G2 F4,E6,E7~E 8 that Theorems about Section 2 gives 2 is the that if G , then N N for for shows~ require (or mod centers) i and 2 do not seem to have been no techniques not known that they will A conjecture as to what H-space for many be useful led to this study by our homotopy in a finite-dimensional questions is the not split Sp(n]/center theory.

I. There is an H-structure on followed by localization. S 7 such that f is an H-map. pf. (Zabrodsky). A separation element argument shows that, allowing different H-structures on S T, the obstruction to f being an H-map lies in Now H 1 4 ( S p ( 2 ) ( 3 ) ) = Z3, and s u b g r o u p o f o r d e r 3. shows ~2 e H14($7 ) g e n e r a t e s A look at the Postnikov system easily f # ( ~ 2 ) ~ 0, t h u s t h e d i s p l a y e d To e o n s t r u c t a cyclic K, c o n s i d e r coset i s 0.

The compact N I of D So L e m m a of L e m m a Theorem by be a i we i. The N and is j u s t the the quotient map. be least in a implies Lie group S m T semisimple tori and let is d e f i n e d tori} T by . Lie group exp: ~T G is (T,S) . map by (See it is w e l l (See, > T is I The C a r t a n - S t i e f e l [i]~ page i01). known that for e x a m p l e [6], need to show plan w i l l then show group identity for NO that I Thus and D Corollary in s set S(N) is the n o r m a l i z e r NO Lie of terms covering is abelian~ determines = S semisimple S N to d e f i n e S(N) component the u n i v e r s a l Since be which connected an e x p r e s s i o n NO G1 ~ G2 two m a x i m a l just torus be ma- i is proved.