THE MULTIPLICATIVE STRUCTURE OF K(n)(BA4)

Let K(n)∗(−) be a Morava K-theory at the prime 2. Invariant theory is used to identify K(n)∗(BA4) as a summand of K(n)∗(BZ/2 × BZ/2). Similarities with H∗(BA4;Z/2) are also discussed.


Introduction
Let G be a finite group, and let N and C denote respectively the normalizer and the centralizer of a p-Sylow subgroup H of G.
For a large family of cohomology theories including the Brown-Peterson cohomology BP * (−) and Morava K-theories K(n) * (−), the author described h * (BG) when H is cyclic [3], and discussed the case "p-rank (H) < 3" in [5], proving in particular that h * (BG) is generated as h * -module by at most two elements if |N : C| divides p − 1.
Results in this paper show that the condition above is really necessary, in fact we have Theorem 0.1.Let K(n) * (−) be a Morava K-theory at the prime 2. K(n) * (BA 4 ) restricts to those elements in which algebraically depend on ).
This paper has several motivations.The knowledge of K(n) * (BA 4 ) could help to have explicit formulae for the K(n) * -Dickson classes.Furthermore, similarities among H * (BA 4 ; Z/2) and K(n) * (BA 4 ) suggest to study K(n) * (BA m ) -whose rank as K(n) * -module can be calculated [6]-to get information on H * (BA m ; Z/2) which is not entirely known for m ≥ 16 (see [1] for the cohomology of several alternating groups).
The author would like to thank the anonymous referee, who drew attention to certain inaccuracies contained in the first version.
In [7], the authors describe H * (P SL 2 F q ) for any odd q: they first calculate the cohomology of the generalized quaternion group Q 2 n+1 of order 2 n+1 , and then use the diagram where D n is the dihedral group of order 2 n , rows are fibrations, and i and j are inclusions of 2-Sylow subgroups.Nevertheless, we show in this section that the special case can be approached in a more direct way.The alternating group A 4 is the central term of the short exact sequence of groups therefore for any mod 2 cohomology theory h * (−), h * (BA 4 ) is isomorphic to the ring of invariants [h * (BV )] Z/3 under the action determined by the map h * (Bφ) induced by an automorphism φ of order 3 in Aut(V ).On Φ is commonly known as norm map.It is easy to see that furthermore the image of Φ restricted to the set of monomials generates [H * (BV )] Z/3 regarded as graded F 2 -vector space.
The invariant of lowest positive degree in This element is actually the Dickson class known in literature as Q 2,1 (see [9]).
The reader will find the relevant invariant theoretic computation in [2] to prove the algebraic dependence of every invariant on Φ(xy), Φ(x 2 y), Φ(xy 2 ).In fact we have the following proposition.Proposition 1.1.As a graded ring, H * (BA 4 ) is isomorphic to and R is the ideal generated by The proposition above can be restated in terms of pure invariant theory.
Corollary 1.2.Suppose that a Z/3-action on F 2 [x, y] is given by x −→ y and y −→ x + y.

The Morava K-theory of BA 4
We recall that Morava K-theory at the prime 2 is a complex oriented cohomology theory with coefficients n ] where deg ν n = −2(2 n − 1), and we have where the Z/3-module structure is defined by the map K(n) * (Bφ), being φ a generator of Z/3 ≤ Aut(V ).The following lemma helps to give a concrete description of the K(n)-invariants.
The element α K (y) is actually the formal sum of x and y with respect to the formal group law of mod 2 Morava K-theory Consider now the norm map Ψ defined as follows: . The map Ψ is obviously the analogue of Φ defined in section 1: it is surjective, and the invariants regarded as F 2 -vector space are spanned by the image of Ψ restricted to monomials.
Notice also that we can equip with a different Z/3-module structure just by posing α H (x) = y and α H (y) = x + y.
Proof: Since K(n) * (−) is 2(2 n − 1)-periodic we can look at classes in K(n) * (BV ) whose degree is between 2 and 2(2 n − 1).In this range, elements of type ν 2 n x h y k are necessarily zero, since either h or k is greater than 2 n .It follows that for any monomial ).
Hence the proposition follows.