How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Question

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has degree 2, and the Koszul signs are imposed (so that in particular $\xi^2 = 0$). As a module over the Steenrod algebra, the only interesting fact is that $x = \beta \xi$, where $\beta$ denotes the Bockstein. The rest of the Steenrod powers can be worked out by hand.

There are two groups of order $p^2$. The $\mathbb F_p$ cohomology of $C_p \times C_p$, including its Steenrod powers, is computable from the Kunneth formula. For the cyclic group $C_{p^2}$, you have to think slightly more, because there is a ring isomorphism $\mathrm{H}^\bullet(C_p, \mathbb F_p) \cong \mathrm{H}^\bullet(C_{p^2}, \mathbb F_p)$, but the Bockstein vanishes on $\mathrm{H}^\bullet(C_{p^2}, \mathbb F_p)$. Still I think the Steenrod algebra action is straightforward to write down.

I want to know about the groups of order $p^3$. The abelian ones are not too hard, I think, and there are two nonabelian groups. The one with exponent $p^2$ is traditionally denoted "$p^{1+2}_-$", and the one with exponent $p$ is traditionally denoted "$p^{1+2}_+$". I care more about the latter one, but I'm happy to hear answers about both. And right now I care most about the prime $p=3$.

The cohomology of these groups was computed in 1968 by Lewis in The Integral Cohomology Rings of Groups of Order $p^3$. Actually, as is clear from the title, Lewis computes the integral cohomology, from which the $\mathbb F_p$-cohomology can be read off using the universal coefficient theorem. For the case I care more about, Lewis finds that $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb Z)$ has the following presentation. (I am quoting from Green, On the cohomology of the sporadic simple group $J_4$, 1993.) The generators are: $$ \begin{matrix} \text{name} & \text{degree} & \text{additive order} \\ \alpha_1, \alpha_2 & 2 & p \\ \nu_1, \nu_2 & 3 & p \\ \theta_j, 2 \leq j \leq p-2 & 2j & p \\ \kappa & 2p-2 & p \\ \zeta & 2p & p^2 \end{matrix}$$ (For the $p=3$ case that I care most about, there are no $\theta$s, since $2 \not\leq 3-2$.) A complete (possibly redundant) list of relations is: $$ \nu_i^2 = 0, \qquad \theta_i^2 = 0, \qquad \alpha_i \theta_j = \nu_i \theta_j = \theta_k \theta_j = \kappa \theta_j = 0$$ $$\alpha_1 \nu_2 = \alpha_2 \nu_1, \qquad \alpha_1 \alpha_2^p = \alpha_2 \alpha_1^p, \qquad \nu_1\alpha_2^p = \nu_2 \alpha_1^p,$$ $$ \alpha_i\kappa = -\alpha_i^p, \qquad \nu_i\kappa = -\alpha_i^{p-1}\nu_i,$$ $$ \kappa^2 = \alpha_1^{2p-2} - \alpha_1^{p-1}\alpha_2^{p-1} + \alpha_2^{2p-2}, $$ $$ \nu_1 \nu_2 = \begin{cases} \theta_3, & p > 3, \\ 3\zeta, & p = 3. \end{cases}$$ From this Green (ibid.), for example, writes down a PBW-type basis.

Question: What is the action of the Steenrod algebra been on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb F_p)$?

I'm not very good at Steenrod algebras. Does the ring structure on the $\mathbb Z$-cohomology suffice to determine the action? For instance, the additive structure of $\mathrm{H}^\bullet(G, \mathbb Z)$ already determines the Bockstein action on $\mathrm{H}^\bullet(G, \mathbb F_p)$. If there is a systematic way to do it, where can I learn to do the computations?

Answer 1

If $P$ is the group of order $p^3$ and exponent $p$, its mod $p$ cohomology ring is known to have its depth = its Krull dimension = rank of a maximal elementary abelian subgroup = 2. A theorem of Jon Carlson then implies that the product of restriction maps $$ H^*(P;\mathbb F_p) \rightarrow \prod_E H^*(E;\mathbb F_p)$$ is monic, where the product is over (conjugacy classes of) subgroups $E \simeq \mathbb Z/p \times \mathbb Z/p$.

Thus the ring you care about, viewed as an algebra equipped with Steenrod operations (I would call this an unstable $A_p$--algebra) embeds in a known unstable algebra. Thus if you know how algebra generators of $H^*(P;\mathbb F_p)$ restrict to the various $H^*(E;\mathbb F_p)$'s, it shouldn't be hard to calculate Steenrod operations.

For example, David Green and Simon King's group cohomology website tells you cohomology ring generators and relations, and these restrictions for the group of order 27. (See https://users.fmi.uni-jena.de/cohomology/27web/27gp3.html) I'll let you take it from here.

[By the way, you began your question with a remark that there isn't so much to the cohomology ring $H^*(C_p;\mathbb F_p)$ as a module over the Steenrod algebra. Yes, it is trivial to compute, but it is a deep theorem, with huge unexpected consequences, that this is a injective object in the category of unstable $A_p$--modules. See the book by Lionel Schwartz on the Sullivan conjecture for more detail.]

Edit the next day: Thanks to Leason for pointing out that my map doesn't detect for $p>3$. To get the detection result, one needs that the depth = Krull dimension, and this turns out to only happen in the one case that I looked up carefully: $p=3$. So in the other cases, the depth will be 1 = rank of the center, and one needs a more general detection theorem, pioneered by Henn-Lannes-Schwartz in the mid 1990's, and then explored by me in various papers about a decade ago. (Totaro later wrote about this in his cohomology book: this is the result mentioned by Heard.) In the case in hand, the range of the detection map for $H^*(P;\mathbb F_p)$ will need one more term in the product: Let $C < P$ be the center: a group of order $p$. The multiplication homomorphism $C \times P \rightarrow P$ induces a map of unstable algebras $$ H^*(P;\mathbb F_p) \rightarrow H^*(C;\mathbb F_p) \otimes H^{*\leq 2p}(P;\mathbb F_p)$$ where the last term means truncate above degree $2p$. That the number $2p$ works to detect all remaining nilpotence is an application of my general result: it can be determined by understanding the restriction to the center.

At any rate, for that original group of order 27 and exponent 3, this isn't needed. (The group of order 27 and exponent 9 will also need that extra factor in the detection map range.)

Answer 2

I think the answer you want can be found in the following article:

AUTHOR = {Leary, I. J.},
 TITLE = {The mod-{$p$} cohomology rings of some {$p$}-groups},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical
          Society},
VOLUME = {112},
  YEAR = {1992},
NUMBER = {1},
 PAGES = {63--75},
  ISSN = {0305-0041},