Assume $X$ is an $H$-space and let's look at homotopy groups of $SP(X)$ where $SP$ here is the infinite symmetric product. By Dold-Thom we have $\pi_i(SP(X))\cong \tilde{H}_i(X)$. Since $X$ is an $H$-space there is a graded product structure on $\tilde{H}_i(X)$ for $i>0$. So this implies that there is a graded product structure on $\pi_i(SP(X))$ for $i>0$. So I was wondering whether it is possible to realize this graded product structure in more geometric way and in the form of a smash product on $SP(X)$? Is there a map $SP(X)\wedge SP(X) \rightarrow SP(X)$ that does this?
To me it seems that we have a map $SP(X)\wedge SP(X) \rightarrow SP(X\wedge X)$, unless we have a smash product on $X$ we won't get a map to $SP(X)$. The smash product on $X$ seems to be something independent of the $H$-space structure.
So another related question is that do nice $H$-spaces also admit a smash product i.e. $X\wedge X \rightarrow X$ where this product somehow depends on the $H$-space structure? (For all smash products let's assume the base point is the identity $e$)
Here is an idea not sure whether it is relevant or not. If $X$ admits a de-loop, $BX$, then the Whitehead product on homotopy groups of $BX$ turns into a graded product structure on the homotopy groups of $X$, not sure whether this is related to the graded product structure coming from Dold-Thom or not.