Define the Steenrod algebra $A^\ast$ to be the algebra of all stable mod $p$ cohomology operations. Without actually computing $A^\ast$, is it possible to see that $A^\ast$ acts faithfully on $H^\ast(BC\_p; \mathbb F\_p)$?


My question is closely related to [this one](https://mathoverflow.net/questions/83096/is-there-a-high-concept-explanation-of-the-dual-steenrod-algebra-as-the-automorp).




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Here is an attempt at an argument. (**EDIT:** As Tyler's comment shows, this argument doesn't work! I'll leave it up, though, as an example of the kind of thing I might hope could be true) Let $\phi: H \to \Sigma^r H$ be a nonzero stable cohomology operation (where $H = H\mathbb F\_p$ is the Eilenberg-MacLane spectrum). Then


$$\phi \wedge 1 : H \wedge H \to \Sigma^r H \wedge H$$


is a nonzero $H$-module map, and so is nonzero on homotopy. We have $H = \varinjlim\_n \Sigma^{-n} K(\mathbb F\_p, n)$ . It follows that


$$\phi \wedge 1: H \wedge \Sigma^{-n} K(\mathbb F\_p, n) \to \Sigma^r H \wedge \Sigma^{-n} K(\mathbb F\_p, n)$$


is nonzero on homotopy for some $n$. I *think* it's the case that $H\_\ast(K(\mathbb F\_p, n))$ is generated under Pontryagin product by $H\_\ast(\Sigma K(\mathbb F\_p, n-1))$ -- but I'm not sure if this is true, much less whether there is a non-computational reason for it. This ought to allow us to induct downwards to show that


$$\phi \wedge 1: H \wedge K(\mathbb F\_p, 1)^N \to \Sigma^r H \wedge K(\mathbb F\_p, 1)^N$$


is nonzero on homotopy for some $N$, which is *almost* the desired conclusion.



Why does the Steenrod algebra act faithfully on $H^\ast(BC_p)$?
As is commented by @Connor Malin, the action of the Steenrod algebra on $H^\*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^\*(B \mathbb{Z}/2)$, since $Sq^{2n+1}x^{2m}=0$ by the Cartan formula. As a matter of fact, the computation of the Hopf ring structure of $H\_\*K(\mathbb{Z}/p,\*)$ shows that for no finite $n$, the action of the Steenrod algebra on $H^\*((B\mathbb{Z}/p)^n)$ can be faithful. One can see this more easily using the last section of T.Kashiwabara [Hopf Rings and Unstable Operations, JPAA 94 (1994) 183-193](https:///core.ac.uk/download/pdf/82509056.pdf), or even from classical computations of Steenrod algebra.


