The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal example, once it is shown that in the path-loop fibration sequence $K(\mathbb{Z},2n) \to P(K(\mathbb{Z},2n+1)) \to K(\mathbb{Z},2n+1)$

1. the fundamental class $v$ of the fiber transgresses to $u$, that of the base

2. this forces a zig-zag of cancellation, up to $v^{p-1}\mapsto u \otimes v^{p-2}$

3. also $v^p$ transgresses to $P^n(u)$, and

4. $u\otimes v^{p-1}$ "transgresses" to $\beta P^n(u)$.

Parts (1), (2) and (3) are easy, but part (4) seems difficult. There is a proof along these lines in a paper of Browder from the mid 1960's (he attributes the proof to Milgram), but the proof of (4) is actually quite hard and leans heavily on algebraic mucking around in the spectral sequence.

Does anyone know of a clever way to prove (4)?