Given a finite subgroup of $G$ sitting inside the Morava stabilizer group $S\_n$, we can form the homotopy fixed point spectrum $E\_n^{hG}$. There is a spectral sequence with $E\_2^{s,t} = H^s(G;\pi\_t(E\_n)) \Rightarrow \pi\_{t-s}(E\_n^{hG})$. 


For resolving further differentials in this spcetral sequence I have seen a way by claiming some elements in the $E\_2$ page is actually the image of certain elements of $\pi\_\*(S)$ under Hurewicz map and then by the known multiplicative relation(for example $h\_1^3 = h\_0^2 h\_2$) we shall know that some elements in this spectral sequence must die and try to figure out who kills it or who it kills. 


**Question**: what is the general way, or is there any general way of proving this kind of claim? i.e. how to prove that something in HFPSS $E\_2$ is actually corresponding to the image of an element from $\pi\_\*(S)$?


I only know some classical results that might be related to this: If we start with $Ext\_{BP\_\*BP}$ and keep track of the element we are interested in inside the chromatic spectral sequence, we will end up with continuous group cohomology of $S\_n$(after tensoring with $\mathbb{F}\_{p^n}$ at some point) thus cohomology of finite subgroup of it. Then we can look at how elements $t\_i\in BP\_\*BP$ works as functions from $S\_n$ to $\mathbb{F}\_{p^n}$ to tell the image of $t\_i$ in group cohomology and finally detecting elements of $\pi\_\*(S)$.


The second question comes from a concrete example of this. By classical result, we know that the non-exist elements $\beta\_{p^j/p^j}$ and their monomials for $p \geq 5$ can be detected in the way above, in group cohomology of a finite group of order $p$ inside $S\_{p-1}$ with coefficient $\mathbb{F}\_{p^{p-1}}$. And my second question is the following:


**Question**: I believe that if I want to rephrase the detection of $\beta\_{p^j/p^j}$ in modern language, it will becomes things like "If $\beta\_{p^j/p^j}$ survives to $\pi\_\*(S)$, then it can be detected in the $C\_p$ homotopy fixed point of $E\_{p-1} $ for $p \geq 5$". So is this true? If so, if I can draw the $E\_2$ of this spectral sequence, could I actually identify elements in $H^\*(C\_p;\mathbb{F}\_{p^{p-1}})$ (where the classical detection happens) inside this $E\_2$? Or is there a suitably defined map from $E\_2$ to $H^\*(C\_p;\mathbb{F}\_{p^{p-1}})$ which maps image of $\beta\_{p^j/p^j}$ to non-zero elements?


Any help or hint is really appreciated.



Computing Homotopy Fixed Point Spectral Sequences related to Morava E thories
In Ravenel's [paper](http://www.math.rochester.edu/people/faculty/doug/mypapers/arf_odd.pdf) on the Arf invariant, he shows (p. 439) that there is a composite of maps
$$
\mathrm{Ext}\_{BP\_\*BP}(BP\_\*,BP\_\*) \to H^\*\_c(\mathbb{S}\_n,E\_\*) \to H^\*(C\_p,E\_\*/\frak{m}),
$$
under which the images of $\alpha\_1,\beta\_1$ and $\beta\_{p,p}$ are non-zero. You can use these to determine the Hurewicz image of classes under the composite
$$
\mathrm{Ext}\_{BP\_\*BP}(BP\_\*,BP\_\*) \to H^\*\_c(\mathbb{S}\_n,E\_\*) \to H^\*(G,E\_\*).
$$


Since this is probably the argument you outline in the first paragraph, it is useful to point out another method, via the functoriality of the Greek letter construction; this is the argument used in Proposition 7 of [this](https://hal.archives-ouvertes.fr/hal-00129678/document) paper by Goerss, Henn, and Mahowald. 


I'm afraid I don't fully understand the second part of the question; if a class survives in the $BP$-Adams spectral sequence, and is detected in the $C\_p$-HFPSS spectral sequence under the map
$$
\mathrm{Ext}\_{BP\_\*BP}(BP\_\*,BP\_\*) \to H^\*(C\_p,E\_\*),
$$
then it must also survive in the latter spectral sequence. 


