The Structure of The Steenrod Algebra
Serre (1953) (for p=2) and Cartan (1954, 1955) (for p>2) described the structure of the Steenrod algebra of stable mod p cohomology operations, showing that it is generated by the Bockstein homomorphism together with the Steenrod reduced powers, and the Adem relations generate the ideal of relations between these generators. In particular they found an explicit basis for the Steenrod algebra. This basis relies on a certain notion of admissibility for integer sequences. We say a sequence
is admissible if for each j, ij ≥ 2ij+1. Then the elements
where I is an admissible sequence, form a basis (the Serre-Cartan basis) for the mod 2 Steenrod algebra. There is a similar basis for the case p > 2 consisting of the elements
such that
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