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 Sn in each degree n>0, with coproduct given by D(Sn) = \sumi+j=n Si \otimes Sj. We introduce a new perspective on the Adem relations in the mod 2 Steenrod algebra A2 by studying the map p\ast dual to the Hopf algebra epimorphism p: F \otimes Z/2 \to A2. We also express Milnor's Hopf algebra conjugation formula in A2\ast in a different form and give a new approach for the conjugation invariant problem in A2\ast.