Let $A\_n$ be the alternating group of $\{1,2,\cdots,n\}$. 


(1). What is the cohomology ring 
$$
H^\*(A\_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?


(2). Are there general results about the cohomology ring 
$$
H^\*(A\_{p+1};\mathbb{Z}/p)
$$
for general primes $p\geq 3$?


(3). What is the cohomology ring 
$$
H^\*(A\_n;\mathbb{Z}\_2)
$$
for $n\geq 4$ (we can impose conditions on $n$)?


Any references?



mod p cohomology ring of alternating groups
An easier way for the first case


Consider $P$, a $3$-Sylow subgroup of $A\_4$. For example, take the cyclic group generated by the cyclic permutation (123). It is self normalizing, so the double coset formula for the compositions $BP\rightarrow BA\_4 \stackrel{tr}{\rightarrow}BP$ where $tr$ denotes the transfer, reduces to the identity. Therefore $BP$ and $BA\_4$ are homotopy equivalent at the prime 3.


For the second case, let's start with the review of the well-known computation of
$H^\*(B\Sigma \_p;Z/p)$. Again as in the above, denote by $P$ the cyclic group generated by the cyclic permutation $(12\cdots p)$. Denote by $N$ the group of affine automorphisms of $Z/p$, considered as a subgroup of the permutation of $\{1,\ldots ,p\}$. Then $N$ is the normalizer of $P$ in $\Sigma \_p$,
with $N/P\cong Gl\_1(Z/p)$ with the canonical action on $Z/p$. Now the double coset formula for the compositions $BP\stackrel{Bi}{\rightarrow} B\Sigma \_p \stackrel{tr}{\rightarrow}BP$ says that in mod $p$ cohomology, this composition is the sum 
of maps induced by the multiplication by $s\in Gl\_1(Z/p)$. In other words,
we have, $$Bi^\*tr^\*(\beta ^{\epsilon}x^i)=\Sigma \_{s\in Gl\_1(Z/p)}\beta ^{\epsilon}s^ix^i\in H^\*(BZ/p,Z/p)\cong Z/p[x]\otimes \Lambda \_{Z/p}(\beta x).$$
The little Theorem of Fermat then shows that this is identity if $p-1\mid i$, $0$ otherwise. Therefore we get
$$H^\*(B\Sigma \_p,Z/p)\cong Z/p[x^{p-1}]\otimes \Lambda \_{Z/p}(\beta x^{p-1})\subset
H^\*(BZ/p,Z/p).$$


Now, if we replace $\Sigma \_p$ with $A\_{p+1}$, what changes is the normalizer.
It is easy to see that $$N\_{A\_{p+1}}(P)=A\_{p+1}\cap \Sigma \_p.$$ We also see that the generator of $Gl\_1(Z/p)$ acts on $P$ as a cyclic permutation of length $p-1$, thus its an odd permutation. This means in the appropriate double coset formula, the only half of $s's$ get involved (squares in $Gl\_1(Z/p)$), and we arrive at the conclusion $$H^\*(BA \_{p+1},Z/p)\cong Z/p[x^\frac{p-1}{2}]\otimes \Lambda \_{Z/p}(\beta x^\frac{p-1}{2})\subset
H^\*(BZ/p,Z/p).$$ When $p=3$, this reduces to the first case treated in the above.


