Abstract
J.H. Harper [1] proved that E8, localized at the prime 5, has the same homotopy type of K(5) X 5(15, 23, 39, 47)X513(5) where #*(5(15,23,39,47); ZJ=A(xls,xZB, x39, *47). See [2] resp. [4] for more details about K(5) resp. 513(5). One would like to know whether 5(15, 23, 39, 47) is irreducible or of the same homotopy type of 57(5)XJ3i9(5). The purpose of this paper is to show that &(x2Z)=x39 where Si is the secondary operation defined by the relation ((l/2)/3P—P/3)(P) +P(/3)=0 and *23, xsg^H*(E8, Z5). This certainly implies that 5(15,23,39,47) is indecomposable. Using this fact one can compute the 5-component of 7r38(E8) which turns out to be zero. In 1970, H. Toda [4] announced that the following cases of mod-/) decompositions of exceptional Lie groups were unknown:
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