Does primitive (resp. to comultiplication) homology classes comes from Hurewicz map?

I heard this statement for loop spaces, but can't find proof for that or counter-example for following question. consider ordinary rational homologies, there are primitive elements with respect to coalgebra structure, it is almost obvious that homology classes released by spheres (with rational coefficient) are such elements, is it all?

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Yes. This is classical, maybe originally in Milnor and Moore's paper on Hopf algebras. For a recent exposition see for example "More concise algebraic topology" by Kate Ponto and myself. If $X$ is a connected $H$-space (say with finitely generated rational homology groups), then the rationalized Hurewicz homomorphism is a monomorphism with image the primitive elements of $H_*(X;\mathbf Q)$. The essential point is that the rationalization of $X$ is equivalent to a product of Eilenberg-MacLane spaces.