If \(A\) is a space, a space \(B\) is said universal for \(A\) if \(B\) is a homotopy commutative, homotopy associative \(H\)-space and there is a map \(A\to B\) which is universal in a natural way. Now localize at an odd prime and take homology with mod-\(p\) coefficients. If \(G\) is a compact, connected, simple Lie group which is torsion free, the author proves the existence of a co-\(H\) space \(A(G)\) such that \(H_*(G)\cong S(\tilde{H}_*A(G))\) with a decomposition \(A(G)\simeq \vee_{i=1}^{p-1} A_i(G)\). One of the main results of the paper is the following one.
Theorem: Suppose \(G\) is one of the following: \(SU(n)\) if \(n\leq (p - 1)(p - 3)\), \(Sp(n)\) if \(2n \leq (p - 1)(p - 3)\), \(Spin(n)\) if \(2n + 1 \leq (p - 1)(p - 3)\); \(G_2\), \(F_4\), or \(E_6\) if \(p \geq 5\); \(E_7\) or \(E_8\) if \(p \geq 7\). Then there is a homotopy decomposition \(G \simeq \prod_{i=1}^{p-1} B_i(G)\) where each \(B_i(G)\) is homotopy associative, homotopy commutative, and universal for \(A_i(G)\).
In some particular cases (for instance, \(SU(n)\) with \(2n<p\)), the previous decomposition is an equivalence of \(H\)-spaces. Applications of these results to the torsion in the homotopy groups of \(G\), including an upper bound on its exponent, are given.