The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie$(n)$, as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups.