Let $G$ be a compact Lie group and let $BG$ be its classifying space. Let $\gamma\colon \Sigma G \to BG$ be the adjoint map for a natural weak equivalence $G \xrightarrow{\sim} \Omega BG$. Taking rational cohomology, it is well-known, that the map $\gamma$ induces an isomorphism between indecomposable elements $QH^*(BG,\mathbb{Q})$ and primitive elements $\mathrm{Prim}\, H^{*-1}(G,\mathbb{Q})$. But I want to know what happens integrally.
To be more precise, the integral cohomology ring modulo torsion $H^*(G,\mathbb{Z})/\mathrm{Tor}$ is still a Hopf algebra isomorphic to an exterior algebra (see Mimura, Toda - Topology of Lie groups, I and II, Chapter VII, Theorem 1.22). Is the homomorphism induced by $\gamma$ still surjective onto primitive elements? If not, what one can say about orders of cokernels?
