An alternative approach to the Adem relations in the mod 2 Steenrod algebra

The Leibniz--Hopf algebra F is the free associative algebra over Z on one generator S^n in each degree n 0, with coproduct given by Delta(S^n) = sum_{i+j=n} S^i otimes S^j. We introduce a new perspective on the Adem relations in the mod 2 Steenrod algebra A_2 by studying the map pi^ast dual to the Hopf algebra epimorphism pi: F otimes Z/2 to A_2. We also express Milnor s Hopf algebra conjugation formula in A_2^ast in a different form and give a new approach for the conjugation invariant problem in A_2^ast.