The Geometry and Cohomology of the Mathieu Group M_{12}

Alejandro Adem, John Maginnis, and R.James Milgram†

§0 INTRODUCTION

It is usually very complex to calculate the cohomology of a finite group G. To date the most significant results have been on the symmetric and alternating groups ([N], [M– M], [Ma], [Mu], [A–M–M]); the general linear groups over a finite field [Q1]; and the extra–special p–groups [Q2].

In the list above only the alternating groups and the GL_{n}( _{2}) are simple. Indeed, among the simple groups only a very few, aside from those above have been understood. It is probable that away from the characteristic, p, H^{∗}(P SL_{n}( _{p^{n}}); /q) is available, but even this is not obvious, especially for q = 2. And certainly we currently have no information on most of the families of groups of Lie type, for example, G_{2}( _{q}), E_{6}( _{q}), E_{7}( _{q}), E_{8}( _{q}), or any of the sporadic groups but M_{11} and J_{1}.

About 15 years ago Quillen introduced very powerful techniques that allowed for a much deeper understanding of group cohomology. However, it is also apparent that successful calculations over /p have only occurred when the group has a well–behaved lattice of p–elementary abelian subgroups. Formidable combinatorial problems arise in the general situation, as well as necessarily complicated multiplicative relations. For example, the general calculation of H^{∗}(GL_{n}( _{p}); /p) remains quite inaccessible, and there has been only marginal progress since Quillen’s landmark results.

In this paper we calculate the mod 2 cohomology of the Mathieu Group M_{12}, one of the 26 sporadic simple groups. It has order 95, 040, and can be represented as a collineation group in P G(5, 3) (the projective space of five dimensions over the field of three elements), leaving invariant a configuration of 12 points. Of the finite simple groups not belonging to infinite families, only M_{11} (of order 7, 920) and J_{1} (of order 175, 560, with elementary abelian 2–Sylow subgroup) have been studied cohomologically ([B–C], [Ch]), both cases fairly immediate calculations. The group M_{12} however, is considerably more complicated, being a rank 3 group with a 2–Sylow subgroup of order 64.

The methods used here combine the classical double coset formula of Cartan–Eilenberg with some of Quillen’s techniques. First we determined the poset space of elementary abelian 2–groups (as well as their normalizers) in M_{12} (see §2 for definitions); next we found that in our list of isotropy subgroups, there were two non–isomorphic groups of order 192. The first centralizes one of the two non-conjugate involutions and one of the three conjugate classes of ( /2)^{3}’s in M_{12}. It is quite well understood. The second normalizes one of the four conjugate classes of /2× /2’s and the second of the three ( /2)^{3}’s. As far as we can find out it first occurs in the literature in [G] in a completely different context, realizing M_{12} as a quotient of the amalgmated free product of the two groups of order 192 above over their intersection. This is obtained by considering automorphisms of trivalent graphs. In §5 we will discuss this approach and, in fact, prove that at the prime p = 2, the cohomology of M_{12} is isomorphic to that of the amalgmated product (5.1).

† Research supported in part by grants to the authors from the N.S.F.

Even from our initial point of view, the double coset structure of W, W′, seemed to contain most of the cohomological information about M_{12}. This became more precise with:

Theorem 3.1: Let H be a 2–Sylow subgroup of M_{12}. There exist two non–isomorphic subgroups W and W′ of order 192 in M_{12} with H as their intersection, such that

H^{∗}(M_{12}; /2) ≅ im (res_{H}^{W})^{∗} ∩ im (res_{H}^{W′})^{∗}

in H^{∗}(H; /2).

Remark: Using a result of [We] together with our explicit analysis of the poset space for M_{12}, and the calculation of the cohomology groups of all of its isotropy groups we obtain the Poincaré series for H^{∗}(M_{12}; /2) as well. It is

p(t) = \frac{(1 + t^{3})^{2}}{(1 − t^{2})(1 − t^{3})(1 − t^{4})}

though this does not reflect the actual ring structure too well. The cohomology ring turns out to be Cohen-Macaulay, i.e. freely and finitely generated over a polynomial subalgebra, in this case a polynomial algebra on generators in dimension 4, 6, and 7 isomorphic to the Dickson algebra H^{∗}(( /2)^{3}; /2)^{GL_{3}( /2)} ⊂ H^{∗}(( /2)^{3}; /2). To reflect this we rewrite the Poincaré series above as

\frac{1 + t^{2} + 3t^{3} + t^{4} + 3t^{5} + 4t^{6} + 2t^{7} + 4t^{8} + 3t^{9} + t^{10} + 3t^{11} + t^{12} + t^{14}}{(1 − t^{4})(1 − t^{6})(1 − t^{7})} .

Remark: The groups G_{2}( _{q}) of order q^{6}(q^{2} − 1)(q^{6} − 1) and ^{3}D_{4}( _{q}) of order q^{12}(q^{2} − 1)(q^{6} − 1)(q^{8} + q^{4} + 1) with q a prime power and q ≡ 3, 5 (mod 8) have the same Sylow 2-subgroups as M_{12}. Consequently, we would expect that the complete analysis we give for H^{∗}(H; /2) will be useful in studying these groups as well.

This work is the first application of the poset space to explicit cohomology calculations in rank larger than 2 (see [We] for a different aspect of this). Geometrically this means that we had to deal with a 2–complex having a large number of cells. In §2 we provide a diagrammatic representation of the orbit space of this complex, including a complete list of isotropy groups. The intersection in H^{∗}(H; /2) is given in §4, where we calculate the cohomology rings H^{∗}(H; /2), H^{∗}(W; /2), and H^{∗}(W′; /2). We then determine the images of (res_{H}^{W})^{∗} and (res_{H}^{W′})^{∗}. Using the program MACAULAY we give the intersection as a ring over the Steenrod algebra A(2) with 8 generators and 14 relations (4.15 - 4.18).

It seems that the combination of techniques exploited here will yield equally tractable results for other sporadic simple groups, in particular we are currently analyzing M_{22} and M_{23} in this way. (For M_{22} see the more detailed comments in §3.)

We include two appendices containing the data on the subgroup structure of M_{12} required for our calculations.

Much of the calculational work here was done on computers. Symbolic manipulation programs like Cayley are not powerful enough to handle a group the size of M_{12}. However, with efficient storage algorithms and good multiplication algorithms, modern machines can be effective for studying groups having orders less than approximately 10^{7}. Indeed, in the case at hand it is likely that most, if not all of the results of §2 could have been obtained by hand, but it was so much easier to use the machines that there appeared to be no choice as to preferred methods.

We would like to thank R. Lee for stimulating our interest in the cohomology of sporadic simple groups as well as David Benson and John Carlson for sharing their insights with us – in particular they suggested that we check for the Cohen-Macaulay condition, and finally David Rusin for invaluable aid with the final determination of the ring structure for H^{∗}(M_{12}; /2) using the program MACAULAY.

§1 M_{12} AND ITS LATTICE OF 2–SYLOW SUBGROUPS

The goal of this section is to provide the classical group–theoretic description of M_{12}, with particular attention to its lattice of 2–Sylow subgroups. A presentation of the Mathieu group M_{12} of order

95040 = 12 ∘ 11 ∘ 10 ∘ 9 ∘ 8 = 2^{6} ∘ 3^{3} ∘ 11 ∘ 5

is given in Hall’s book [H], p.80 ; see also [S], p.286. It has generators

\begin{matrix}u_{1} : & (1, 2, 3)(4, 5, 6)(7, 8, 9) \\ u_{2} : & (1, 4, 7)(2, 5, 8)(3, 6, 9) \\ a : & (2, 4, 3, 7)(5, 6, 9, 8) \\ b : & (2, 5, 3, 9)(4, 8, 7, 6) \\ x : & (1, 10)(4, 5)(6, 8)(7, 9) \\ y : & (1, 11)(4, 6)(5, 9)(7, 8) \\ z : & (1, 12)(4, 7)(5, 6)(8, 9) \end{matrix}

and the first six elements generate M_{11}, the Mathieu group of order 7920 = 11∘10∘9∘8 = 2^{4} ∘ 3^{3} ∘ 11 ∘ 5. These are both simple sporadic groups and Benson and Carlson [B–C] have recently determined the cohomology groups of M_{11}. However, M_{12} seems considerably harder. We have, to begin (see also [W]): Theorem 1.1: The elements a, b, generate a copy of the quaternion group Q_{8}. The elements x, y, z generate a copy of S_{4} = Aut(Q_{8}), and the subgroup

W = < a, b, x, y, z >⊂ M_{12} is Q_{8} ×_{T} Aut(Q_{8}),

the semi-direct product (often called the holomorph of Q_{8} in the literature). Proof: First, verifying that < a, b > = Q_{8} is easy. Second we have

\begin{aligned}xax & = (2, 5, 3, 9)(4, 8, 7, 6) = b so xbx = a. \\ yay & = (2, 6, 3, 8)(9, 4, 5, 7) = ba so yby = b^{−1}. \\ zbz & = (2, 6, 3, 8)(7, 9, 4, 5) = ab so zaz = a^{−1}. \end{aligned}

so < x, y, z > normalizes Q_{8}. But, also

yz = (1, 11, 12)(4, 5, 8)(6, 7, 9)

has order 3, while

xyz = (1, 10, 11, 12)(4, 8, 7, 6)

has order 4. It follows that < x, y, z > is a quotient of the group

S_{4} = {A, B | A^{2} = B^{3} = (AB)^{4} = 1}.

On the other hand, projection onto the subgroup S_{4} ⊂ S_{12} with generators 1, 10, 11, 12, is a surjection, so < x, y, z > = S_{4}, as claimed. The result now follows.

The index of W in M_{12} is 495, and W ∩ M_{11} is a group of order 3 ∘ 16, and has index 165 in M_{11}. W has order 192, and contains 3 2-Sylow subgroups of M_{12}. In §2 we will see that from the geometry of the poset space of elementary abelian subgroups, there arises a distinct subgroup W′ of order 192. In §3 we will see that W, W′ play a strikingly dominant role in our cohomology calculations.

We will use the particular 2-Sylow subgroup H ⊂ W ⊂ M_{12}. It is generated by the three elements b, z, and zz in the list below. Moreover, the quotient by the commutator subgroup is ( /2)^{3} so there is a decomposition of H into 8 cosets of H′_{2}, with coset generators as follows

coset element  representation
#1  id
#2  b = (2 5 3 9)(4 8 7 6)
#3  zz = (1 11)(4 7)(6 8)(10 12)
#4  zz ∘ b = (1 11)(2 5 3 9)(4 6 7 8)(10 12)
#5  z = (1 12)(4 7)(5 6)(8 9)
#6  z ∘ b = (1 12)(2 5 4 6 3 9 7 8)
#7  z ∘ zz = (1 10 12 11)(5 8 9 6)
#8 z ∘ zz ∘ b = (1 10 12 11)(2 5 7 6 3 9 4 8)

To obtain the commutator subgroup explicitly we check that the commutator of z and zz is (1 12)(5 9)(6 8)(10 11), while the commutator of z and b is (2 7 3 4)(5 8 9 6). These two elements generate a group ( /2 × /4) which has order 8, and is consequently the subgroup H′.

There are two special maximal subgroups of H. The first is the subgroup consisting of the first four cosets above. We denote it H_{21}. It can also be written as the semi-direct product

Q_{8} ×_{α} ( /2 × /2)

where the action is by inner automorphism. It can be described as follows: in M_{12}, let W be the centralizer of b^{2}. Then |W | = 192, and it contains 3 conjugates of H. The intersection of any two is H_{21}.

The second maximal subgroup consists of the first, third, fifth, and seventh cosets, and we denote it H_{22}. One can check that

H_{22} = H ∩ αHα^{−1}

where α = (1 6 7)(2 11 5)(3 10 9)6(4 12 8) so H_{22} is also a maximal subgroup which is given as the intersection of H with a conjugate subgroup in M_{12}. Theorem 1.2: The two subgoups H_{21} and H_{22} are the only maximal subgroups of H which are obtained as the intersections of H and a second 2-Sylow subgroup of M_{12}. Moreover, besides these two groups there are only two other non-commutative subgroups, both of order 16 and contained in H_{22} which arise as the intersection of H with a second 2-Sylow subgroup in M_{12}.

H_{21} /

H  G_{1}  L_{1} = ( /2)^{3}
\  /

H_{22} \

G_{2}  L_{2} = ( /2)^{3}

The intersections of order ≥ 8

Proof: This is a computer enumeration. We began by finding and storing generators for the 1485 cosets of H is M_{12}. Then a direct calculation of the double cosets of H in M_{12} was generated. For details, see Appendix 1.

Remark: We enumerate the various intersections of the groups above:

(1)  H_{21} ∩ H_{22} ≅ D_{8} × /2

(2)  G_{2} ∩ H_{21} = G_{1} ∩ H_{21} ≅ ( /2)^{3}

(3)  L_{2} ∩ H_{21} = L_{1} ∩ H_{21} = ( /2)^{2} ⊂ H′.

Using the computations in Appendix 1, we describe the conjugacy classes of elementary abelian 2–groups in M_{12}. Theorem 1.3: In H there are 5 conjugacy classes of groups of the form (Z/2)^{3}. In M_{12} these become exactly three conjugacy classes. Representatives for the conjugacy classes in H are the groups

\begin{aligned}K = & < a^{2}, zz, [z, zz] >, \\ L = & < a^{2}, [z, zz], b[z, zz] >, \\ M = & < a^{2}, b[z, zz]a, zz ∘ a >, \\ K_{1} = & < a^{2}, [z, zz], z >, \\ L_{1} = & < a^{2}, [z, zz], zz ∘ a > . \end{aligned}

In M_{12}, L and L_{1} are conjugate, as are K and K_{1}. In particular each conjugacy class of maximal elementary 2-group in M_{12} is represented as a subgroup of H_{21}.

Proof: This is a direct enumeration, considerably aided by computer calculations. First, there are 7 conjugacy classes of elements of order 2 in H. They are given by the table

Class  cycle form  number of conjugates
a^{2}  (2,3)(4,7)(5,9)(6,8)  1
[z, zz]  (1,12)(5,9)(6,8)(10,11)  2
b[z, zz]  (1,12)(2,9)(3,5)(4,6)(7,8)(10,11)  4
zz  (1,11)(4,7)(6,8)(10,12)  4
zz ∘ a  (1,11)(2,7)(3,4)(5,8)(6,9)(10,12)  4
zz ∘ b ∘ a  (1,11)(2,8)(3,6)(4,5)(7,9)(10,12)  4
z  (1,12)(4,7)(5,6)(8,9)  8

Next the centralizers (in H) of each of the elements above except the first were constructed. The centralizer of the last was a copy of ( /2)^{3}, while the centralizers of the remaining elements all copies of /2 × D_{8} with the center generated by the element and a^{2} in all cases. Also, in each case the element generated the /2 split summand. Each of these groups has exactly 2 subgroups ( /2)^{3}, and these were enumerated. The resulting table, of necessity, had all the conjugacy classes of such subgroups of H represented. The table follows

Representative  Group  Type
b ∘ [z, zz]  < a^{2}, zz, b ∘ [z, zz] >  4^{3}6^{4}
b ∘ [z, zz]  < a^{2}, zz ∘ b ∘ a, b ∘ [z, zz] >  4^{1}6^{6}
zz  < a^{2}, [z, zz], zz >  4^{7}
zz ∘ a  < a^{2}, [z, zz], zz ∘ a >  4^{3}6^{4}
zz ∘ a  < a^{2}, zz ∘ a, b ∘ [z, zz] ∘ a >  4^{1}6^{6}
zz ∘ b ∘ a  < a^{2}, zz ∘ [z, zz], zz ∘ b ∘ a >  4^{3}6^{4}
z  < a^{2}, z, [z, zz] >  4^{7}

Here, the type symbol 4^{3}6^{4} denotes the fact that of the seven non-identity elements of the group 3 have a cycle decomposition as a product of 4 transpositions in the embedding into S_{12}, while 4 have a cycle decomposition as a product of 6 transpositions.

In the table above, the first group and the sixth are easily seen to be conjugate in H, as are the second and fifth. It is also directly verified that these are the only possibilities. For example, in the fourth group all four of the elements of cycle type 6 are conjugate in H, while this is not true in the first group. Similarly, the seventh group is the centralizer of one of its elements in H, while the third is not. Thus the first statement in the theorem is verified.

The element (1, 6, 12, 8)(5, 10, 9, 11) ∈ M_{12} conjugates the third group to the seventh, while the element (1, 12, 11)(4, 8, 5)(6, 9, 7) ∈ M_{12} conjugates the fourth group above to the first.

Theorem 1.4: There are 4 conjugacy clases of 2-groups isomorphic to ( /2)^{2} in M_{12}. Representatives may be given as

R = < a^{2}, zz > Type 4^{3}
R_{1} = < zz, [z, zz] > Type 4^{3}
S = < a^{2}, b[z, zz] > Type 4^{1}6^{2}
T = < zz · b · a, b[z, zz] > Type 6^{3}

Proof: We first need to observe the result Lemma 1.5: Every subgroup G ⊂ H which is isomorphic to /2 × /2 is contained in an maximal elementary subgroup X = ( /2)^{3}. Proof: If a^{2} ∈ G then G may be assumed to contain one of the other 6 elements in the table of conjugacy class representatives above. The second table gives an extremal group containing G. If a^{2} ∉ G then < G, a^{2} >= ( /2)^{3} will serve.

Continuing with the proof of the theorem, assume first that G contains an element α of type (2, 2, 2, 2), then there is a β ∈ M_{12} so βαβ^{−1} = a^{2} and consequently βGβ^{−1} ⊂ H. Similarly, if G just consists of the identity and three elements of type (2, 2, 2, 2, 2, 2) then it is conjugate to a subgroup of the group M above. In the first case it is either conjugate to the subgroup < a^{2}, [z, zz] > in L, one of the 6 subgroups of K containing a^{2}, or one of the 6 subgroups of M containing a^{2}.

In the case of the subgroup of M, note that the elements of type (2, 2, 2, 2, 2, 2) form a single orbit under the action of the automorphism group. Consequently, any two subgroups containing a^{2} are conjugate. It is also easy, using the particular form of the action of S_{4} on the 6 elements of this type in M to show that any two subgroups consisting only of elements of this type and the identity are conjugate. Similarly, if the group is conjugate to a subgroup of L which contains a^{2} and also contains an element of type (2, 2, 2, 2, 2, 2), then there is a Sylow 2-subgroup of S_{4}, which fixes a^{2}, but acts transitively on the 4 elements of this type in L. It follows again that any two subgroups of this type are conjugate. Finally, it is easy to see that the intersection of L and M is a subgroup of this type. So we have shown there are unique conjugacy classes of ( /2)^{2} having the types of S and T.

It remains to discuss those groups G containing 3 elements of type (2, 2, 2, 2). The discussion above shows that we can assume they are contained in L. The action of the group N_{M_{12}}(L) on L decomposes these 7 subgroups into two orbits, the first containing R, and the second R_{1}. To verify that, in fact, R and R_{1} are not conjugate in M_{12}, we used a computer program to compare all the conjugacy classes of R to R_{1}.

The only remaining thing to discuss is the way in which these groups occur in the intersections with conjugates of H in M_{12}. This was checked on the computer, and the result was Theorem 1.6: Only the conjugacy classes of T =< zz · b · a, b[z, zz] > among the groups isomorphic to ( /2)^{2} in H occur as the intersections, H ∩ αHα^{−1}, of H with one of its conjugates in M_{12}.

This completes our discussion of the lattice of subgroups of H in M_{12}.

§2 THE POSET SPACE FOR M_{12}

Let X be a partially ordered set (poset), and denote by |X| the simplicial complex associated to it. This is the simplicial complex whose vertices are the elements of X, and whose simplices are the non-empty finite chains in X.

Now if G is a finite group, Brown [B] and later Quillen [Q] introduced the study of the complexes associated to the posets

S_{p}(G) = poset of non-identity p-subgroups of G

A_{p}(G) = poset of non-identity p-elementary abelian subgroups of G.

The following proposition summarizes the main properties of these poset spaces (see [Q] for details)

Proposition 2.1: Let G be a finite group of p-rank r. Then

(a) A_{p}(G) is a simplicial complex of dimension r-1, with an action of G induced by conjugation.

(b) the natural inclusion A_{p}(G) ↪ S_{p}(G) is a homotopy equivalence

(c) if G contains a non-trivial normal p-subgroup, then A_{p}(G) is contractible, and (d)

Ĥ_{G}^{∗}(A_{p}(G), F_{p}) ≅ Ĥ^{∗}(G, F_{p}).

Remark: here Ĥ_{G}^{∗}(X) denotes the equivariant Tate cohomology of X, see [B] for a description. Part (d) is a particular case of a result for discrete groups proved by Brown ([B], pg. 293). Proof: The first three parts of 2.1 are explicit in [Q]. For part (d) we observe that if P ⊂ G is a p-Sylow subgroup, then the singular set of the P -action on A_{p}(G) is contractible ([Q], 4.1). Denote this singular set by S_{p}(|A_{p}(G)|).

We will show that the dual of the augmentation _{p} ⟶^{ϵ^{∗}}C^{∗}(|A_{p}|) induces an isomorphism in Tate cohomology. It fits into a short exact sequence

0⟶ _{p} ⟶^{ϵ^{∗}}C^{∗}(|A_{p}|)⟶C̃^{∗}(|A_{p}|)⟶0.

Consequently, it reduces to showing that Ĥ_{G}^{∗}(C̃^{∗}(|A_{p}|)) ≡ 0. By the usual transfer restriction argument it suffices to show that Ĥ_{P}^{∗} (C̃^{∗}(|A|_{p})) ≡ 0 for P a p-Sylow subgroup of G. Hence we can assume that G = P and prove that ϵ_{P}^{∗} is an isomorphism. However, we have a commutative diagram with the vertical arrow an isomorphism:

Ĥ^{∗}(P; _{p}) ⟶^{ϵ_{p}^{∗}}  Ĥ^{∗}(|A_{p}(G)|)
↘ ϵ̄^{∗}  ↓i_{p}^{∗}

Ĥ_{P}^{∗} (S_{P} (|A_{p}(G)|)).

Using the fact that S_{P} (|A_{p}(G)|) is contractible and a spectral sequence argument we deduce that ϵ̄^{∗} must also be an equivalence, completing the proof.

It is now clear from 2.1 that for cohomological considerations, the poset space is potentially most useful for groups without normal p-subgroups, e.g. simple groups. It also indicates that H^{∗}(G) can be approached using equivariant cohomology. Consequently it is important to understand the singular set of the G–action on A_{p}(G), as well as the isotropy subgroups, which in this case are the normalizers of the p-elementary abelian subgroups and the flags associated to them. For M_{12} we will show that in fact two isotropy subgroups of order 192 completely control the cohomology at the prime 2. We refer to [We] for other aspects of this point of view.

We now proceed to describe A_{2}(M_{12}). We start by providing a complete list of isotropy subgroups occurring for this two dimensional complex.

Proposition 2.2: The two conjugacy classes of involutions 4^{1}, and 6^{1} in M_{12} satisfy a. The centralizer of 4^{1} is W, the holomorph of Q_{8}. b. The centralizer of 6^{1} is /2 × S_{5}. Proof: (a) has already been discussed and is well known. (b) is a computer calculation, though it was known previously [W] that |N_{M_{12}}(6^{1})| = 240. First, the elements of the centralizer of (6^{1}) were listed where the specific element used was

(1, 11)(2, 8)(3, 6)(4, 5)(7, 9)(10, 12) = x.

Then 2 elements α, β ∈ N_{M_{12}}(6^{1}) were found with α^{2} = β^{3} = (αβ)^{5} = 1, so < α, β >= A_{5}. Indeed, α = a^{2} = (2, 3)(4, 7)(5, 9)(6, 8), β = (1, 10, 4)(2, 7, 3)(5, 11, 12)(6, 8, 9) are suitable choices. Clearly, x ∉ A_{5}, so | < x, α, β > | = 120, the resulting group is normal in N_{M_{12}}(< x >), and is given as a central extension

/2 =< x > ⟶^{◁} < x, α, β > ⟶A_{5}.

But there is only one (non-trivial) central extension of A_{5} by /2 since H^{2}(A_{5}; /2) = /2, and this is the binary icosahedral group, Ã_{5}. However, the Sylow 2-subgroup of Ã_{5} is Q_{8}, the quaternion group. Consequently, if we had the non-trivial extension we would have that x = y^{2} for some y, but an inspection of the elements of N_{M_{12}}(< x >) shows that this is impossible so < x, α, β >= /2 × A_{5}. This implies the existence of a normal extension

/2 × A_{5} ⟶^{◁}N_{M_{12}}(< x >)⟶ /2.

Now, the number of extensions of /2 × A_{5} by /2 with a given action map

/2⟶Out( /2 × A_{5}),

is given by H^{2}( /2; ( /2 × A_{5})). The center of /2 × A_{5} is /2 =< x >, so this cohomology group is /2. Also, Out( /2×A_{5}) = /2, so it follows that there are precisely 4 distinct extensions of the above type, the product, /2 × /2 × A_{5}, the group /2 × S_{5} and two extensions in which x = y^{2}. But the product has a subgroup ( /2)^{4} which is not a subgroup of M_{12} and we have already seen that x is not a square in N_{M_{12}}(< x >). The proposition follows.

For later use we need the following Corollary 2.3: In N_{M_{12}}(< x >) there are precisely 5 conjugacy classes of involutions: x, α, xα, a class t, which represents the involution (1, 2) ∈ S_{5}, i.e. a class which extends A_{5} to S_{5} and xt. The orbit of α under conjugation in /2 × S_{5} contains 15 elements, and these are the only elements in this group of type 4^{1}. (This is clear except for the last statment. This was verified by checking that there were exactly 15 elements in N_{M_{12}}(< x >) of type 4^{1} by just inspecting the table of elements.)

Next we study the normalizer of the first conjugacy class of groups ( /2)^{2}, those of type 4_{I}^{2}, i.e. conjugate to < a^{2}, zz >. We have Theorem 2.4: The normalizer of the group < a^{2}, zz > in M_{12}, N_{M_{12}}(4_{I}^{3}), has order 192 but is not isomorphic to W. More exactly, the centralizer of < a^{2}, zz >, (4_{I}^{3}) is given by

\begin{aligned}_{M_{12}}(< a^{2}, zz >) & = ( /4 × /4) ×_{T} /2 \\ & = \{a, d, f | a^{4} = d^{4} = f^{2} = 1; ad = da, faf = a^{−1}, fdf = d^{−1}\}, \end{aligned}

and N_{M_{12}}(4_{I}^{3}) is given as a semi-direct product

M_{12}(< a^{2}, zz >) ×_{α} S_{3}

with the element of order 3 acting by T (a) = d, T (d) = (ad)^{−1}, T (f) = a^{3}df, while an element g of order 2 acts as g(f) = a^{2}f, g(a) = a^{−1}, g(d) = ad. Proof: Set

\begin{aligned}a & = (2, 5, 3, 9)(4, 8, 7, 6), \\ d & = (1, 12, 11, 10)(4, 6, 7, 8), \\ f & = (1, 10)(4, 7)(5, 9)(11, 12), \\ T & = (1, 4, 2)(3, 11, 7)(5, 10, 6)(8, 9, 12), \\ g & = (1, 11)(2, 7)(3, 4)(5, 8)(6, 9)(10, 12). \end{aligned}

It is now systematic to verify all the claims above, except perhaps the claim that

N_{M_{12}}(4_{I}^{3}) ≠ W.

But note that a 2-Sylow subgroup of N_{M_{12}}(4_{I}^{3}) ⊂ W since f and g centralize a^{2}. In particular, if N_{M_{12}}(4_{I}^{3}) has a center it must be < a^{2} >, but T does not centralize a, so N_{M_{12}}(4_{I}^{3}) has trivial center and cannot be isomorphic to W.

The next normalizer that we identify is N_{M_{12}}(4^{3}6^{4}). We found the next result somewhat unexpected. Proposition 2.5: N_{M_{12}}(4_{I}^{3}) = N_{M_{12}}(4^{3}6^{4}). Proof: We can check that fa = (1, 10)(2, 5)(3, 9)(4, 6)(7, 8)(11, 12). Consequently

N_{M_{12}}(< a^{2}, zz >) contains the subgroup < a^{2}, zz, fa >

which has type 4^{3}6^{4}. Moreover, T (fa) = a^{3}f = fa, and g(fa) = a^{2}fa^{3} = fa. Also, dfad^{−1} = zzfa, and af = a^{2}fa. It follows that the group N_{M_{12}}(4_{I}^{3}) ⊂ N_{M_{12}}(4^{3}6^{4}). On the other hand, (4^{3}6^{4}) ⊂ (4_{I}^{3}), and N_{M_{12}}(4^{3}6^{4})/ _{M_{12}}(4^{3}6^{4}) = S_{4}. The result follows. Remark: The normalizer of < a^{2}, fa > is contained in W since this group has type 4^{1}6^{2}, so every element in N_{M_{12}}(< a^{2}, fa >) must also fix a^{2}. From this it is easy to check that |N_{M_{12}}(< a^{2}, fa >)| = 32. Indeed, the group is generated by d^{2}, g, a, and f. It is an extra special two group

/2 =< a^{2} > ⟶^{◁}N_{M_{12}}(< a^{2}, fa >)⟶( /2)^{4}

where the latter group has generators a, d^{2}, g, and f. The extension data is given by [a, d^{2}] = [d^{2}, f] = 1, [f, g] = [g, d^{2}] = [a, g] = [a, f] = a^{2}. Remark: Similarly, the group N_{M_{12}}(4^{1}6^{6}) ⊂ W, and its order is directly seen to be 96. Indeed, checking we find that a representative group is < a^{2}, fa, g >⊂ W. Both d^{2}, and a normalize this group, but they are independent in their action. Moreover, as we have seen the normalizer contains an element of order 3. Consequently we have that the 2-Sylow subgroup of N_{M_{12}}(< a^{2}, fa, g >) is isomorphic to the group N_{M_{12}}(< a^{2}, fa >) described above. Moreover we have the group given as a normal extension

( /2)^{3} ⟶^{◁}N_{M_{12}}(4^{1}6^{6})⟶A_{4}.

The next group to consider is N_{M_{12}}(4_{II}^{3}) ≅ N_{M_{12}}(< zz, [z, zz] >). A computer check shows that this group has order 48 and is contained in W. In particular we have two ways of describing the resulting group,

( /2)^{3} =< a^{2}, zz, [z, zz] > ⟶^{◁}N_{M_{12}}(< zz, [z, zz] >)⟶S_{3},

and

< a^{2} >= /2⟶^{◁}N_{M_{12}}(< zz.[z.zz] >)⟶S_{4}.

It remains to discuss the normalizer N_{M_{12}}(6^{3}). Since N_{M_{12}}(6^{1}) = /2 × S_{5} with a class of type 6^{1} generating the /2 summand, it suffices to check the centralizer of an element of order 2 and type 6^{1} in the S_{5} piece. But this is given by S_{3} × /2 since, as we saw above, all the elements of type 4^{1} ∈ N_{M_{12}}(< x >) are contained in A_{5}. Hence, the centralizer of 6^{3} has order 24 and is contained in N_{M_{12}}(6^{1}). From the last section Out_{M_{12}}(6^{3}) = /3 so |N_{M_{12}}(6^{3})| = 72 and the group is given as the product

N_{M_{12}}(6^{3}) ≅ S_{3} × A_{4}.

Next we analyze the flags in the 2-elementary subgroups of M_{12}

Definition 2.6: Let G be a group, then a p-elementary k-flag in G is a sequence subgroups G_{i} ⊂ G and proper inclusions

G_{1} ⊂ G_{2} ⊂ ⋯ ⊂ G_{k}

with each G_{i} an elementary p-group.

Two k-flags (G_{1} ⊂ ⋯ ⊂ G_{k}), (G′_{1} ⊂ ⋯ ⊂ G′_{k}) are conjugate if there is a g ∈ G with gG_{i}g^{−1} = G′_{i}, i = 1, 2, …, k. The isotropy group in G of the k-flag (G_{1} ⊂ ⋯ ⊂ G_{k}) is the set of all g ∈ G with gG_{i}g^{−1} = G_{i}, i = 1, 2, …, k. Note that the isotropy group of a flag is the intersection of the normalizers of the G_{i}.

In this section we study only the 2-elementary k-flags in M_{12} so we suppress the modifiers M_{12} and 2-elementary in what follows. Proposition 2.7: For flags of the form /2 ⊂ ( /2)^{2} the conjugacy classes of the groups G_{1}, G_{2} determine the conjugacy class of the flag. Consequently, we have the following table

Type  isotropy group  order
(4^{1}, 4_{I}^{3})  H  64
(4^{1}, 4_{II}^{3})  D_{8} × /2 =< a^{2}, z, zz >  16
(4^{1}, 4^{1}6^{2})  N_{M_{12}}(4^{1}6^{2})  32
(6^{1}, 4^{1}6^{2})  _{M_{12}}(4^{1}6^{2})  16
(6^{1}, 6^{3})  S_{3} × ( /2)^{2}  24

(This is direct from the results of the last section.) Proposition 2.8: For 2-flags of the form ( /2)^{2} ⊂ ( /2)^{3} we also have that the conjugacy classes of the two groups G_{1}, G_{2} determine the conjugacy class of the flag. Consequently we have the following table

Type  isotropy group  order
(4_{I}^{3}, 4^{7})  H  64
(4_{II}^{3}, 4^{7})  N_{M_{12}}(4_{II}^{3})  48
(4_{I}^{3}, 4^{3}6^{4})  N_{M_{12}}(4^{3}6^{4})  192
(4^{1}6^{2}, 4^{3}6^{4})  N_{M_{12}}(4^{1}6^{2})  32
(4^{1}6^{2}, 4^{1}6^{6})  N_{M_{12}}(4^{1}6^{2})  32
(6^{3}, 4^{1}6^{6})  /2 × A_{4}  24

There are two exceptional cases for 2-flags of type /2 ⊂ ( /2)^{3}. Specifically we have Theorem 2.9: There are two distinct conjugacy classes of 2-flags of the form (4^{1}, 4^{7}). The first is represented by (< a^{2} >, < a^{2}, zz, [z, zz] >) and has isotropy group W, while the second is represented by (< zz >, < a^{2}, zz, [z, zz] >) and has isotropy group _{M_{12}}(< a^{2}, zz >) = H_{2,2} of order 32. Proof: That there are two distinct isotropy groups is clear since W = N_{M_{12}}(4^{7}) fixes a^{2} and makes a single orbit of the remaining terms. Then the isotropy group of (< zz >, < a^{2}, zz, [z, zz] >) certainly fixes both zz and a^{2} and consequently is contained in M_{12}(< a^{2}, zz >) = ( /4)^{2} ×_{T} /2. On the other hand, this centralizer, being contained in W, normalizes < a^{2}, zz, [z, zz] >.

Finally, note that there is an element θ ∈ M_{12} with θa^{2}θ^{−1} = zz. Then θW θ^{−1} ∩W = M_{12}(< a^{2}, zz >), and it follows that there is a choice of Sylow 2-subgroup H′ ⊂ W with H′ ∩ θH′θ^{−1} also equal to this centralizer. Now, since _{M_{12}}(< a^{2}, zz >) ≅ H̸_{2,1} it must be H_{2,2}.

Otherwise we have Proposition 2.10: Except for the 2 cases above a 2-flag of the form /2 ⊂ ( /2)^{3} is determined up to conjugacy by the conjugacy types of the groups G_{1}, G_{2}. Consequently, we have the table

Type  isotropy group  order
(4^{1}, 4^{3}6^{4})  H  64
(6^{1}, 4^{3}6^{4})  /2 × S_{4}  48
(4^{1}, 4^{1}6^{6})  N_{M_{12}}(4^{1}6^{6})  96
(6^{1}, 4^{1}6^{6})  /2 × D_{8}  16

Finally we consider the 3-flags. From our previous discussion we have directly Theorem 2.11: There are 9 conjugacy classes of 3-flags /2 ⊂ ( /2)^{2} ⊂ ( /2)^{3} in M_{12}. There are two of type (4^{1}, 4_{I}^{3}, 4^{7}) represented by (< a^{2} >, < a^{2}, zz >, < a^{2}, zz, [z, zz] >) with isotropy group H, and (< zz >, < a^{2}, zz >, < a^{2}, zz, [z, zz] >) with isotropy group H_{2,2}. The remaining classes are determined by the types of the subgroups so we have the table

Type  isotropy group  order
(4^{1}, 4_{II}^{3}, 4^{7})  D_{8} × /2  16
(4^{1}, 4_{I}^{3}, 4^{3}6^{4})  H  64
(4^{1}, 4^{1}6^{2}, 4^{3}6^{4})  N_{M_{12}}(4^{1}6^{2})  32
(4^{1}, 4^{1}6^{2}, 4^{1}6^{6})  N_{M_{12}}(4^{1}6^{2})  32
(6^{1}, 4^{1}6^{2}, 4^{3}6^{4})  _{M_{12}}(4^{1}6^{2})  16
(6^{1}, 4^{1}6^{2}, 4^{1}6^{6})  _{M_{12}}(4^{1}6^{2})  16
(6^{1}, 6^{3}, 4^{1}6^{6})  ( /2)^{3}  8

On the following page we give a schematic representation of the orbit space for |A_{2}(M_{12})|, with isotropy subgroups labeled as before. One can check that χ(|A_{2}(M_{12})|) = 3201, necessarily congruent to one modulo 64 (the order of the 2-Sylow subgroup). Given the explicit nature of our data, it might by of some interest to analyze the representations H_{i}(|A_{2}(M_{12})|, _{2}), i = 1, 2 and verify whether or not they are projective _{2}(M_{12}) modules as prior evidence suggests. (See [We] for more on this.)

For readers interested in the purely computational aspects of determining the subgroups of M_{12}, we refer to [B-R] where a complete list of conjugacy classes of subgroups is given.

§3. THE MAIN RESULT

There are two non-isomorphic groups of order 192 contained in M_{12} as we have seen in the last section. The first, W, centralizes an involution of type (2, 2, 2, 2) while the second, W′, is the normalizer of a subgroup ( /2)^{3} of type 4^{3}6^{4} and of < a^{2}, zz >. We may assume W, W′ choosen so that W ∩ W′ = H, our usual 2-Sylow subgroup of M_{12}. Theorem 3.1: The cohomology ring H^{∗}(M_{12}; /2) is the intersection of the images

(res_{H}^{W})^{∗} : H^{∗}(W; /2) ↪ H^{∗}(H; /2)

and

(res_{H}^{W′})^{∗} : H^{∗}(W′; /2) ↪ H^{∗}(H; /2).

Proof: A key theorem of Cartan-Eilenberg, the double coset formula [C-E, p. 259], identifies H^{∗}(G; /p) as the set of stable elements in H^{∗}(G_{p}; /p) where G_{p} is a p-Sylow subgroup and stable means the following: let g_{1}, …, g_{r} be double coset generators for G, so G = ∐ G_{p}g_{i}G_{p}, and define G_{g_{i}} = G_{p} ∩ g_{i}G_{p}g_{i}^{−1}. Then Ḡ_{g_{i}} = g_{i}^{−1}G_{g_{i}}g_{i} ⊂ G_{p} and we have that any element ϵ ∈ im(res_{G_{p}}^{G})^{∗} must satisfy

3.2  (i_{g})^{∗}(res_{G_{g_{i}}}^{G_{p}})^{∗}(α) = (resG_{p} ; Ḡ_{g_{i}})^{∗}(α)

where i_{g} : Ḡ_{g_{i}}⟶G_{g_{i}} is conjugation by g_{i}. Conversely, an element α ∈ H^{∗}(G_{p}; /p) is stable if it satisfies 3.2 for each double coset representative g_{i}.

If G_{p} ⊂ L ⊂ G and ḡ_{i} are double coset representatives for L so G = ∐ Lḡ_{i}L, then first, H^{∗}(G; /p)⟶H^{∗}(L; /p) is an injection since L contains a copy of G_{p}, and second, an element in H^{∗}(L; /p) is in the image of H^{∗}(G; /p) if and only if it is stable with respect to the ḡ_{i}, see e.g. [C–E, Prop. 9.4, p. 259]. That is, we need only check the subgroups Lḡ_{i} = L ∩ ḡ_{i}Lḡ_{i}^{−1}.

In our case G = M_{12} and the group L will be W. Appendix 2 gives a computer generated table listing generators for the 10 non-trivial double cosets of W, (there are 11 including W itself), as well as the intersection group W_{g_{i}} and W̄_{g_{i}} next to it. They are arranged so that elements on the same line are conjugate by g_{i}. Often, it will turn out that the particular choice of generator given by the computer is inconvenient to our argument. Where that happens we also list the conjugates (in W) of the group W̄_{g_{i}} which are obtained using different choices for g_{i} to generate the same double coset.

We now consider the elements coset by coset. The first element, (1, 2, 3)(4, 5, 6)(7, 8, 9) acts as the identity on W_{g_{1}}. Hence, i_{g_{1}} = id and 3.2 gives no restriction. The second element g_{2} = (1, 3, 11, 2)(4, 6, 8, 7)(5, 9)(10, 12) has W_{g_{2}} = W̄_{g_{2}} conjugate in W to K_{1}. But the element (1, 4, 2)(3, 11, 7)(5, 10, 6)(8, 9, 12) ∈ W′ takes K to a conjugate of K_{1} in W′. It follows that on the intersection of the two restriction maps in 3.1, the stability condition generated by g_{2} on K_{1} is equivalent to a condition for K, where i_{g_{2}} represents the effect of an automorphism of K. But W is the normalizer of K, hence, on the image of H^{∗}(W; /2) in H^{∗}(H; /2), the stability condition is automatically satisfied.

The third element has i_{g_{3}} = id so there is no condition. The fourth generator gives Ḡ_{g_{4}} ≠ G_{g_{4}}, but the 9^{th} conjugate of Ḡ_{g_{4}} does equal G_{g_{4}}. Hence, we can replace the generator by one which takes G_{g_{4}} to itself. This subgroup is < a^{2}[z, zz], z >. In W′ it is conjugate to a subgroup of K, which in W is conjugate to either < a^{2}, zz > or < zz, [z, zz] >. But W′ normalizes the first, while the normalizer of the second is contained in W. In either case, the stability condition is automatically satisfied on the intersection in 3.1.

The fifth and sixth generators both have W_{g_{i}} conjugate to W̄_{g_{i}}. Indeed, for the fifth, the tenth conjugate of W̄_{g_{5}} is W_{g_{5}}, while for the sixth we use the 22^{nd} conjugate.

For g_{7} the intersection is the dihedral group D_{6}. Since i_{g} takes the first element to itself, and it generates the 2-Sylow subgroup of the intersection, there is no condition.

For g_{8} the story is the same as that for g_{4}. For g_{9}, note that W_{g_{9}} ≅ A_{4}, the alternating group on 4 letters. Now replace W̄_{g_{9}} by the third conjugate, and note that i_{g} on the Sylow 2-subgroup is the identity. Hence, there is no condition.

Finally, we consider the 10^{th} double coset generator. Here W_{g_{10}} = H_{22}, and replacing W̄_{g_{10}} by the third conjugate, we see that conjugation by the new g_{10} gives an automorphism of H_{22}. In particular, it acts as an automorphism on the center of H_{22} which is < a^{2}, zz >. But this means that g_{10} ∈ W′ since W′ is the normalizer of < a^{2}, zz >, and once more, on the intersection of the two images, the stability condition is automatically satisfied.

Thus we have shown that every element in the intersection is in the image of the cohomology groups H^{∗}(M_{12}; /2) in H^{∗}(H; /2). But, by naturality,

im(res_{H}^{M_{12}})^{∗} ⊂ im(res_{H}^{W})^{∗} ∩ im(res_{H}^{W′})^{∗},

and 3.1 follows.

Remark: There is a similar decomposition for M_{22}. There are two subgroups K, S in M_{22} with < K, S >= M_{22}, and K ∩ S contains a 2-Sylow subgroup. However, the analogue of 3.1 is not true in this case. The 2 Sylow subgroup H of M_{22} contains exactly two subgroups ( /2)^{4}. Both are normal in H, but the normalizer of the first (in M_{22}) is a semi-direct product ( /2)^{4} ×_{α} A_{6} = K while the normalizer for the second is a semi-direct product ( /2)^{4} ×_{β} S_{5} = S. See e.g. [J] for further details.

A computer check shows that there are elements v ∈ S, w so that M_{22} = K ∪ KvK ∪ KwK. However, w ∉ S, and K ∩ wKw^{−1} = A_{6} is normalized by w so that the action restricts to one of the Sylow 2 subgroups of A_{6} as a non-trivial outer automorphism. From this it follows that the containment H^{∗}(M_{22}; /2) ⊂ H^{∗}(S; /2) ∩ H^{∗}(K; /2) is proper! A complete discussion will appear in a sequel.

In §5 we will see how our result relates the cohomology of M_{12} to that of the amalgamated product W ∗_{H} W′.

§4 THE CALCULATION OF H^{∗}(M_{12}; /2)

In this section we use the results in §3 to calculate H^{∗}(M_{12}; /2). The procedure is to first study the cohomology rings of the groups H_{22}, H_{21}, use these to obtain the ring H^{∗}(H; /2), then use the double coset formula to obtain H^{∗}(W; /2), H^{∗}(W′; /2). The images of these groups are next explicitly calculated in H^{∗}(H; /2) and the generators of the low dimensional intersections are determined. The group H_{22}

We begin by calculating the cohomology of H_{22}. The result is Theorem 4.1: There are no nilpotent elements in H^{∗}(H_{22}; /2). Indeed we have

\begin{aligned}H^{∗}(H_{22}; /2) & ≅ _{2}[r, s] ⊗ _{2}[a, b, c]/\{a^{2} + ac, b^{2} + bc\} \\ & ≅ _{2}[r, s, c](1, a, b, ab) \end{aligned}

where r, s are both 2-dimensional, while a, b, and c are 1-dimensional. Proof: The center of H_{22} is /2 × /2 with generators b^{2} and m = [z, zz]. Thus it can be given as a central extension

/2 × /2⟶H_{22}⟶( /2)^{3}

and the K invariants are seen to be a^{2} + ac, b^{2} + bc, since we have that the generators for the quotient ( /2)^{3} are the images of l = [z, b], z, and zz while [l, z] = l^{2} = b^{2}, [zz, l] = l^{2}, m = [z, zz], and z^{2} = zz^{2} = 1.

Next, a routine calculation shows that in the resulting spectral sequence E_{3} = E_{∞}. From this the result above is immediate.

From the above, an alternate description of H_{22} is as the semi-direct product

/4 × /4 ×_{T} /2

where the generators of the /4 × /4 are l, l · z · zz, and z generates the twisting /2.

The group H_{21} We now turn our attention to H_{21}, which can be described as a semi-direct product

Q_{8} ×_{α} (Z/2)^{2},

where the action is inner automorphism. An alternate description is as the central extension

Z/2 ◁ H⟶(Z/2)^{4}

with κ-invariant a^{2} + ab + b^{2} + ac + bd ∈ H^{∗}((Z/2)^{4}; /2) = F_{2}[a, b, c, d].

This second description shows that H_{21} is an extra special 2-group (see [Q2]), since the matrix

( \begin{matrix}1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} )

is clearly non-singular mod 2. Consequently, the spectral sequence of the extension above collapses at E_{4}, and Quillen’s structure theorem for the cohomology of such groups shows that

Theorem 4.2:

H^{∗}(H_{21}; /2) = F_{2}[κ_{4}][a, b, c, d]/(a^{2} + ab + b^{2} + ac + bd, a^{2}b + ab^{2} + a^{2}c + ac^{2} + b^{2}d + bd^{2}).

Here κ_{4} is the 4^{th} power of the 1-dimensional generator on the fiber. However, the description above does not give us sufficient control, so we compare the calculation above with the description coming from the spectral sequence of the extension

Q_{8} ◁ H_{21}⟶(Z/2)^{2}.

Here, E_{2} = H^{∗}(( /2)^{2}; /2) ⊗ H^{∗}(Q_{8}; /2) since the action of ( /2)^{2} on Q_{8} is via inner automorphisms, and these act trivially on passing to homology. A quick check, using the previous calculation and the corresponding dimension counts in the first 4 dimensions shows that here E_{2} = E_{∞}.

Thus we have an additive description of H^{∗}(H_{21}; /2) as

F_{2}[κ_{4}, c, d](1, a, b, a^{2}, b^{2}, a^{2}b).

That is to say, it is a free module over F_{2}[κ_{4}, c, d] with 6 generators, as given above. However, the multiplicative structure is quite twisted. The relations are those given above. But note the particular consequences

a^{3} = (ac^{2} + a^{2}d + acd), b^{3} = (bd^{2} + b^{2}c + bcd),
a^{2}b + b^{2}a = a^{2}c + ac^{2} + b^{2}d + bd^{2}, ab = a^{2} + b^{2} + ac + bd.

The group H

We calculate H^{∗}(H; /2) now, using the spectral sequence associated to the (split) extension sequence

1⟶H_{21}⟶H⟶ /2⟶1

which allows us to write H as a semi-direct product H_{21} ×_{T} /2. The E_{2}^{ij} term is H^{i}( /2; {H^{j}(H_{21}; /2)}) where the coefficients H^{j}(H_{21}; /2) are twisted by the action of /2 given as follows a ↔ b, c ↔ d, κ_{4} fixed.

These cohomology groups are described as follows. First E_{2}^{0j} = H^{j}(H_{21}; /2)^{/2} , and, for i > 0 let H̃^{j} = H^{j}(H_{21}; /2)^{/2} /im(1 + T^{∗}), then E_{2}^{ij} = H̃^{j} ∪ e^{i} where e is the non-trivial element in E_{2}^{10} = H^{1}( /2; /2).

Explicitly, M = _{2}[k_{4}, c, d](1, a, b, a^{2}, b^{2}) is a _{2}[ /2]-submodule of H^{∗}(H_{21}; /2). M splits as a direct sum of _{2}[ /2] submodules

M ≃ _{2}[k_{4}, c, d] ⊕ _{2}[k_{4}, c, d](a, b) ⊕ _{2}[k_{4}, c, d](a^{2}, b^{2}).

The quotient module is N = H^{∗}(H_{21}; /2)/M ≃ _{2}[k_{4}, c, d](a^{2}b). Lemma 4.3: M^{/2} = _{2}[k_{4}, c + d, cd](1, a + b, a^{2} + b^{2}, ab, a^{2}b + ab^{2}). Proof: We introduce the notation s_{1} = b, t_{1} = c + d, s_{2} = ab, t_{2} = cd. Clearly 2[k_{4}, c, d]^{/2} = _{2}[k_{4}, t_{1}, t_{2}]. Moreover, we can write _{2}[k_{4}, c, d] = _{2}[k_{4}, t_{1}, t_{2}](1, c), and from this we see that 1, a + b, ac + bd, a^{2} + b^{2}, a^{2}c + b^{2}d generate M^{/2} freely over 1[k_{4}, t_{1}, t_{2}]. Now, note that ab = (a + b)^{2} + (ac + bd). Also,

a^{2}c + b^{2}d = (ac + bd)(a + b) + (c + d)(ab)
= (a^{2} + b^{2} + ab)(a + b) + (c + d)(ab)
= s_{1}^{3} + s_{1}s_{2} + t_{1}s_{2}
= s_{1}^{2}t_{1} + s_{1}t_{2} + s_{1}s_{2} + t_{1}s_{2}.

Now, consider the exact sequence of _{2}[ /2]-modules

1⟶M⟶H^{∗}(H_{21}; /2)⟶N ⟶1

which induces a long exact sequence in cohomology

1⟶M^{/2} ⟶H^{∗}(H_{21}; /2)^{/2} ⟶N^{/2} ⟶^{δ^{0}}H^{1}( /2; M)⟶H^{1}( /2; H^{∗}(H_{21}; /2))⟶ · · · .

The action of the generator T of /2 on the quotient module N is given by

\begin{aligned}T (a^{2}b) & = ab^{2} = a^{2}b + a^{2}c + ac^{2} + b^{2}d + bd^{2} \\ & ≅ a^{2}b (mod M). \end{aligned}

Thus N^{/2} ≃ _{2}[k_{4}, c + d, cd](a^{2}b). Also

δ^{0}(a^{2}b) = a^{2}b + ab^{2} = a^{2}c + ac^{2} + b^{2}d + bd^{2}

which is (1+T)(a^{2}c+ac^{2}) and hence represents 0 in H^{1}( /2; M). It follows that the class

L_{3} = a^{2}b + a^{2}c + ac^{2}

is invariant under T in H^{3}(H_{21}; /2), and maps onto the generator of N^{/2} . We have proved the first part of Theorem 4.4: H^{∗}(H_{21}; /2)^{/2} is generated as an _{2}-algebra by s_{1} = a + b, t_{1} = c + d, k_{4}, s_{2} = ab, t_{2} = cd, and L_{3} = a^{2}b + a^{2}c + ac^{2}. As a module over _{2}[t_{1}, t_{2}, k_{4}], it is free on 1, s_{1}, s_{1}^{2}, s_{2}, s_{1}s_{2}, and L. Therefore, the following is a complete list of relations for H^{∗}(H_{21}; /2)^{/2} :

\begin{aligned}s_{1}^{3} & = s_{1}^{2}t_{1} + s_{1}t_{2} \\ s_{2}^{2} & = s_{1}s_{2}t_{1} + s_{2}t_{1}^{2} \\ s_{1}^{2}s_{2} & = t_{1}L + s_{2}t_{1}^{2} + s_{2}t_{2} \\ s_{1}L & = t_{1}L + s_{1}s_{2}t_{1} + s_{2}t_{1}^{2} + s_{2}t_{2} \\ s_{2}L & = s_{1}s_{2}t_{1}^{2} + s_{2}t_{1}t_{2} + s_{2}t_{1}^{3} \\ L^{2} & = t_{1}t_{2}L + s_{1}s_{2}t_{1}^{3} + s_{2}t_{1}^{2}t_{2} + s_{2}t_{1}^{4}. \end{aligned}

Proof: The relations were found using MAXIMA by applying the relations between a, b, c, d already described in H^{∗}(H_{21}; /2). The rest of the theorem follows from the discussion preceeding it.

We now turn to the calculation of H^{∗}(H; /2) itself. Theorem 4.5: H^{∗}(H; /2) is generated by classes denoted e_{1}, s_{1}, t_{1}, s_{2}, t_{2}, L_{3} and k_{4}. We can, in fact, write H^{∗}(H; /2) as follows

H^{∗}(H; /2) ≃ _{2}[k_{4}, t_{1}, t_{2}](1, s_{1}, s_{1}^{2}, s_{2}, s_{1}s_{2}, L) ⊕ e_{1} ∘ _{2}[e_{1}, t_{2}, k_{4}](1, L).

Note: Upon restricting to H^{∗}(H_{21}; /2), e_{1} ↦ 0 and the other classes restrict to the classes

of the same names in the previous theorem. Proof: The claim is really that the spectral sequence of the extension

1⟶H_{21}⟶H⟶ /2⟶1

collapses. The quotient /2 splits, which detects the generator e_{1}. E_{2}^{0,∗} was calculated in the previous theorem, and noting the traces

\begin{aligned}(1 + T)(a) & = a + b = s_{1} \\ (1 + T)(c) & = c + d = t_{1} \\ (1 + T)(a^{2} + ac) & = a^{2} + b^{2} + ac + bd = ab = s_{2}, \end{aligned}

we find that H^{1}( /2; H^{∗}(H_{21}; /2)) is generated by t_{2}, L_{3}, and k_{4}. Indeed, it is isomorphic to e_{1} ∪ A where A ≅ _{2}[k_{4}, t_{2}](1, L_{3}), and H^{i}( /2; H^{∗}(H_{21}; /2)) ≅ e_{1}^{i} _{2}[k_{4}, t_{2}](1, L_{3}). From this it follows that the only possible differentials are determined by their values on k_{4}. But the class k_{4} is the Stiefel-Whitney class of a representation of H, and so it must be an infinite cycle.

Note that we have filtered relations s_{1}e_{1} ≅ t_{1}e_{1} ≅ s_{2}t_{2} ≅ 0. Corollary 4.6: The Poincaré series for H^{∗}(H; /2) has the form

P (H) = \frac{1}{(1 − t)^{3}} .

Proof: From the expression for H^{∗}(H; /2) in the theorem above we can write

P (H) = \frac{1 + t + 2t^{2} + 2t^{3}}{(1 − t)(1 − t^{2})(1 − t^{4})} + \frac{t + t^{4}}{(1 − t)(1 − t^{2})(1 − t^{4})}

but this is

\frac{1 + 2t + 2t^{2} + 2t^{3} + t^{4}}{(1 − t)(1 − t^{2})(1 − t^{4})} = \frac{(1 + t + t^{2} + t^{3})(1 + t)}{(1 − t)(1 − t^{2})(1 − t^{4})} = \frac{1}{(1 − t)^{3}} .

Recall that H^{∗}(H; /2) is generated by classes e_{1}, s_{1}, t_{1}, s_{2}, t_{2}, L_{3} and k_{4}, and we know the relations up to a multiple of e_{1} (since we know the restrictions to H^{∗}(H_{21}; /2)):

4.7  \begin{matrix}(1) & s_{1}e_{1} = s_{2}e_{1} = t_{1}e_{1} = 0 \\ (2) & s_{1}^{3} = s_{1}^{2}t_{1} + s_{1}t_{2} \\ (3) & s_{2}^{2} = s_{1}s_{2}t_{1} + s_{2}t_{1}^{2} \\ (4) & s_{1}^{2}s_{2} = t_{1}L + s_{2}t_{1}^{2} + s_{2}t_{2} \\ (5) & s_{1}L = t_{1}L + s_{1}s_{2}t_{1} + s_{2}t_{1}^{2} + s_{2}t_{2} \\ (6) & s_{2}L = s_{1}s_{2}t_{1}^{2} + s_{2}t_{1}t_{2} + s_{2}t_{1}^{3} \\ (7) & L^{2} = t_{1}t_{2}L + s_{1}s_{2}t_{1}^{3} + s_{2}t_{1}^{2}t_{2} + s_{2}t_{1}^{4} \end{matrix}

To detect any possible missing e_{1}-factors we would like to compute the restriction map to H^{∗}(H_{22}; /2). The following are straightforeward:

\begin{matrix}s_{1} & ↦ 0 \\ e_{1} & ↦ b \\ t_{1} & ↦ b + c \\ s_{2} & ↦ a(b + c) \\ t_{2} & ↦ s \\ k_{4} & ↦ r^{2} + rs. \end{matrix}

Thus the relations (1), (2), and (3) are valid. The only difficulty is that the restriction to H^{∗}(H_{22}; /2) of the class L is not at all clear. The class L is determined up to a multiple of e_{1} by res_{H_{21}}^{H}(L) = a^{2}b + a^{2}c + ac^{2}.

To go further we consider the Gysin sequence for the extension

1⟶H_{22}⟶H⟶ /2⟶1,

⟶H^{∗−1}(H; /2)⟶^{∪s_{1}}H^{∗}(H; /2)⟶^{res}H^{∗}(H_{22}; /2)⟶^{tr^{∗}}H^{∗}(H; /2)⟶.

From the restrictions calculated above, we see that neither the class (a) nor the class (r) is in the image of restriction, so the transfer map is non-zero on each. In fact we must have

\begin{aligned}tr^{∗}(a) & = e_{1} \\ tr^{∗}(b) & = t_{2} + s_{1}^{2} + s_{1}t_{1} \end{aligned}

since these are the only available classes in the kernel of cupping with s_{1} in H^{∗}(H; /2). Thus we also find that

\begin{aligned}tr^{∗}(br) & = e_{1}(t_{2} + s_{1}^{2} + s_{1}t_{1}) = t_{2}e_{1} \\ tr^{∗}(as) & = e_{1}t_{2}. \end{aligned}

So tr^{∗}(br + as) = 0 and br + as must be in the image of restriction. The class L must be involved, so that

res_{H_{22}}^{H}(L) = br + as + χ

with χ ∈ span < b, c, ab + ac, s >.

We can discover more about the class χ by considering the restriction to H^{∗}(H_{22}; /2) of the expression

s_{1}^{2}s_{2} + t_{1}L + s_{2}t_{1}^{2} + s_{2}t_{2}.

We find that res_{H_{22}}^{H}(s_{1}^{2}s_{2} + t_{1}L + s_{2}t_{1}^{2} + s_{2}t_{2}) = (b + c)χ + abc^{2} + ac^{3}. Since this expression is a relation in H^{∗}(H_{21}; /2) the restriction to H^{∗}(H_{22}; /2) must be a multiple of e_{1}. Thus

χ = a(b + c)c + bY

and so res_{H_{22}}^{H}(L) = br + as + ac^{2} + abc + bY. But recall that L was only determined up to a multiple of e_{1} so we may assume Y = 0 and

res_{H_{22}}^{H}(L) = br + as + ac^{2} + abc.

We now find that the relations (4), (5), and (6) of 4.7 are all valid as stated since their restrictions to H^{∗}(H_{22}; /2) are zero, but that

res_{H_{22}}^{H}(L^{2} + t_{1}t_{2}L + s_{1}s_{2}t_{1}^{3} + s_{2}t_{1}^{2}t_{2} + s_{2}t_{1}^{4})
= b^{2}r^{2} + abs^{2} = b^{2}(r^{2} + rs) + bs(as + br + ac^{2} + abc)
= res_{H_{22}}^{H}(e_{1}^{2}k_{4} + e_{1}t_{2}L).

so we have shown Theorem 4.8: H^{∗}(H; /2) = _{2}[e_{1}, s_{1}, t_{1}, s_{2}, t_{2}, L_{3}, k_{4}]/R where R is the set of relations

\begin{aligned}s_{1}e_{1} & = s_{2}e_{1} = t_{1}e_{1} = 0 \\ s_{1}^{3} & = s_{1}^{2}t_{1} + s_{1}t_{2} \\ s_{2}^{2} & = s_{1}s_{2}t_{1} + s_{2}t_{1}^{2} \\ s_{1}^{2}s_{2} & = t_{1}L + s_{2}t_{1}^{2} + s_{2}t_{2} \\ s_{1}L & = t_{1}L + s_{1}s_{2}t_{1} + s_{2}t_{1}^{2} + s_{2}t_{2} \\ s_{2}L & = s_{1}s_{2}t_{1}^{2} + s_{2}t_{1}t_{2} + s_{2}t_{1}^{3} \\ L^{2} & = t_{1}t_{2}L + s_{1}s_{2}t_{1}^{3} + s_{2}t_{1}^{2}t_{2} + s_{2}t_{1}^{4} + e_{1}^{2}k_{4} + e_{1}t_{2}L. \end{aligned}

Note: We have squaring operations of the form Sq^{1}(s_{2}) = s_{1}s_{2}, Sq^{1}(t_{2}) = (e_{1} + t_{1})t_{2}, Sq^{1}(L) = s_{1}s_{2}t_{1} + s_{2}t_{1}^{2}, Sq^{2}(L) = e_{1}k_{4} + (t_{2} + e_{1}^{2})L + t_{1}^{2}L + s_{2}t_{1}t_{2}, Sq^{1}(k_{4}) = 0,

Sq^{2}(k_{4}) = (t_{2} + e_{1}^{2} + t_{1}^{2})k_{4}

Corollary 4.9: The nilpotent elements of H^{∗}(H; /2) are all contained in the ideal (s_{1}).

More exactly, by using the relation D_{8}∗D_{8} ≅ Q_{8}∗Q_{8}, and the fact that H_{21} ≅ Q_{8}∗Q_{8}, it is not hard to see that H^{∗}(H_{21}; /2) actually has no nilpotents in its cohomology ring. Consequently, the same is true for H^{∗}(H; /2).

The group W

We now turn to the group H^{∗}(W; /2). Since H_{21} is normal in W it follows from the double coset formula that H^{∗}(W; /2) = (res_{H}^{H_{21}})^{−1} (H^{∗}(H_{21}; /2)^{S_{3}}). Hence, we begin by obtaining H^{∗}(H_{21}; /2)^{S_{3}}.

Note that H^{∗}(H_{21}; /2)^{S_{3}} ⊂ H^{∗}(H_{21}; /2)^{/2} , and writing Σ_{τ} = 1 + τ + τ^{2} where τ generates the normal /3 ⊂ S_{3}, we have that T Σ_{τ} = Σ_{τ} T, so

Σ_{τ} H^{∗}(H_{21}; /2)^{/2} = H^{∗}(H_{21}; /2)^{S_{3}}.

To obtain this algebra explicitly, note first that _{2}[k_{4}, t_{1}, t_{2}]^{Σ_{3}} = _{2}[k_{4}, t_{2} + t_{1}^{2}, t_{1}t_{2}]. Moreover

2[k_{4}, t_{1}, t_{2}] = _{2}[k_{4}, t_{1}, t_{2}]^{Σ_{3}}(1, t_{1}, t_{1}^{2}).

consequently, H^{∗}(H_{21}; /2)^{/2} is free over _{2}[k_{4}, t_{2} + t_{1}^{2}, t_{1}t_{2}] with the 18 generators

{1, t_{1}, t_{1}^{2}} × {1, s_{1}, s_{1}^{2}, s_{2}, s_{1}s_{2}, L}.

Let us calculate some traces:

Σ(s_{1}) = (a + b) + (a) + (b) = 0
Σ(s_{1}^{2}) = 0
Σ(s_{2}) = (ab) + b(a + b) + (a + b)a = a^{2} + ab + b^{2}
= s_{2} + s_{1}^{2}
Σ(s_{1}s_{2}) = s_{1}s_{2}
Σ(L) = (a^{2}b + a^{2}c + ac^{2}) + (b^{2}(a + b) + b^{2}(c + d) + b(c + d)^{2})
+ ((a + b)^{2}a + (a + b)^{2}d + (a + b)d^{2})
= L + a^{3} + b^{3} + a^{2}d + b^{2}c + ad^{2} + bc^{2}
= L + s_{1}^{3} + s_{1}^{2}t_{1} + s_{1}t_{1}^{2}
= L + s_{1}t_{2} + s_{1}t_{1}^{2}.

We must also calculate

Σ(c) = (c + d) + (d) + (c) = 0
Σ(c^{2}) = 0
Σ(s_{1}t_{1}) = ac + bd = s_{2} + s_{1}^{2}
Σ(s_{1}^{2}t_{1}) = (a + b)^{2}(c + d) + a^{2}d + b^{2}c = a^{2}c + b^{2}d
= s_{1}^{2}t_{1} + s_{1}s_{2} + s_{1}t_{2} + s_{2}t_{1}
Σ(s_{1}t_{1}^{2}) = (a + b)(c + d)^{2} + ad^{2} + bc^{2} = ac^{2} + bd^{2}
= s_{1}^{2}t_{1} + s_{1}t_{2} + s_{2}t_{1}
Σ(s_{1}^{2}t_{1}^{2}) = s_{2}^{2} + s_{1}^{4}

Finally, we can use the invariance of s_{2} + s_{1}^{2}, s_{1}s_{2}, and L + s_{1}t_{2} + s_{1}t_{1}^{2} to conclude that

Σ(s_{2}t_{1}) = Σ(s_{1}^{2}t_{1})
Σ(s_{2}t_{1}^{2}) = Σ(s_{1}^{2}t_{1}^{2})
Σ(s_{1}s_{2}t_{1}) = Σ(s_{1}s_{2}t_{1}^{2}) = 0
Σ(Lt_{1}) = Σ(s_{1}t_{1}t_{2} + s_{1}t_{1}^{3}) = (t_{2} + t_{1}^{2}) · (s_{2} + s_{1}^{2})
Σ(Lt_{1}^{2}) = Σ(s_{1}t_{1}^{2}t_{2} + s_{1}t_{1}^{4}) = (t_{2} + t_{1}^{2}) · Σ(s_{1}t_{1}^{2}).

Thus H^{∗}(H_{21}; /2)^{Σ_{3}} is free as a module over _{2}[k_{4}, t_{2} + t_{1}^{2}, t_{1}t_{2}] with six generators

{1, s_{2} + s_{1}^{2}, s_{1}s_{2}, L + s_{1}t_{2} + s_{1}t_{1}^{2}, s_{2}t_{1} + s_{1}t_{2} + s_{1}^{2}t_{1}, s_{2}^{2} + s_{1}^{4}}

It now follows that Theorem 4.10:

\begin{aligned}H^{∗}(W; /2) & = _{2}[k_{4}, t_{2} + t_{1}^{2}, t_{1}t_{2}]{(1, s_{2} + s_{1}^{2}, s_{1}s_{2}, L + s_{1}t_{2} + s_{1}t_{1}^{2}, \atop s_{1}t_{2} + s_{2}t_{1} + s_{1}^{2}t_{1}, t_{2}^{2} + t_{1}^{4})} \\ & ⊕ e_{1} · _{2}[e_{1}, k_{4}, t_{2} + t_{1}^{2}](1, L + s_{1}t_{2} + s_{1}t_{1}^{2}). \end{aligned}

with Poincaré series

\frac{1 + t + 2t^{2} + 4t^{3} + 2t^{4} + t^{5} + t^{6}}{(1 − t^{2})(1 − t^{3})(1 − t^{4}).}

The group W′

As was the case with W, we have that H^{∗}(W′; /2) = (res_{H_{22}}^{H})^{−1} H^{∗}(H_{22}; /2)^{S_{3}}, where now, the /3 action is given by the formulae of 2.4. The description of H^{∗}(H_{22}; /2) given in 4.2 is not entirely precise as to the identity of the elements r, s. However, in homology an element γ of order 3 in the normalizer of H_{22} in W′ acts on duals of r, s by γ(r̄) = s̄, γ(s̄) = r̄ + s̄. (Here, the bars denote homology duals.) Actually, once we have choosen r̄ then the action of γ serves to determine s̄. Similarly in cohomology we can assume γ(r) = s, γ(s) = r + s. In dimension 1 the situation is clearer and we have γ(a) = b + c, γ(b) = a + b + c, and γ(c) = c. Similarly, there is an element θ of order 2 in the normalizer which acts by θ(r) = r + s, θ(s) = θγ(r) = γ^{2}θ(r) = s, while in dimension 1 this same element acts by θ(a) = a + b, θ(b) = b, θ(c) = c.

\begin{aligned}γ : & a ↦ b + c \\ & b ↦ a + b + c \\ & c ↦ c \\ & r ↦ s ↦ r + s \end{aligned}  \begin{aligned}θ : & a ↔ a + b \\ & r ↔ r + s \\ & fixes b, c, s. \end{aligned}

To calculate the Σ_{3} invariants we begin by studying the /2 invariants. Write

H^{∗}(H_{22}; /2) = _{2}[r, s, c] ⊕ _{2}[r, s, c](a, b) ⊕ _{2}[r, s, c](ab)

and denote by M the submodule, over _{2}[ /2]

M = _{2}[r, s, c] ⊕ _{2}[r, s, c](a, b).

Then the quotient module N = H^{∗}(H_{22}; /2)/M is

N ≃ _{2}[r, s, c](ab).

The exact sequence

0⟶M⟶H^{∗}(H_{22}; /2)⟶N ⟶0

induces a long exact sequence in cohomology

0⟶M^{/2} ⟶ (H^{∗}(H_{22}; /2))^{/2} ⟶N^{/2} ⟶^{δ}H^{1}( /2; M)⟶ · · · .

Lemma 4.11: M^{/2} = _{2}[c, s, r^{2} + rs](1, b, as + br). Lemma 4.12: N^{/2} ≃ _{2}[c, s, r^{2} + rs](ab) and δ(ab) = 0. Proof: θ(ab) = (a + b)b = ab + b^{2} = ab + bc ≡ ab mod M. δ(ab) = (1 + θ)(ab) = b^{2} = bc = (1 + θ)(ac) = 0 ∈ H^{1}( /2; M). Theorem 4.13:

H^{∗}(H_{22}; /2)^{/2} = _{2}[c, s, r^{2} + rs](1, b, as + br, ab + ac).

Proof: Clear from the above. Note that (ab + ac) represents the term (ab) ∈ N^{/2} .

Noting the _{2}[c, s, r^{2} + rs]^{Σ_{3}} = _{2}[c, r^{2} + rs + s^{2}, r^{2}s + rs^{2}] and _{2}[c, s, f^{2} + rs] = 2[c, s, r^{2} + rs]^{Σ_{3}}(1, s, s^{2}), we need to consider the traces (1 + γ + γ^{2}) on the twelve generators

{1, s, s^{2}} ∘ {1, b, as + br, ab + ac}.

We find, in particular

\begin{aligned}(a + γ + γ^{2})(ab + ac) & = (ab + ac) \\ (1 + γ + γ^{2})(as + br) & = as + cs + br \\ (1 + γ + grg^{2})(bs^{2}) & = br^{2} + as^{2} + cs^{2}, \end{aligned}

all the others being either 0 or repetitions of those above. Consequently, we have

H^{∗}(H_{22}; /2)^{Σ_{3}} = _{2}[c, r^{2} + rs + s^{2}, r^{2}s + rs^{2}](1, ab + ac, as + cs + br, as^{2} + cs^{2} + br^{2}).

It now follows that Theorem 4.14:

\begin{aligned}H^{∗}(W′; /2) = & _{2}[e_{1} + t_{1}, k_{4} + t_{2}^{2}, t_{2}k_{4}]{(1, s_{2}, L + e_{1}t_{2} + t_{1}t_{2} + t_{1}s_{2}, \atop t_{2}L + e_{1}k_{4} + e_{1}t_{2}^{2} + t_{1}t_{2}^{2} + t_{1}s_{2}t_{2})} \\ & ⊕ _{2}[e_{1} + t_{1}, t_{2}, k_{4} + t_{2}^{2}](s_{1}, s_{1}^{2}, s_{1}s_{2}, s_{1}^{2}s_{2}) \end{aligned}

with Poincaré series

\frac{1 + t + 2t^{2} + 3t^{3} + 2t^{4} + 3t^{5} + 2t^{6} + t^{7} + t^{8}}{(1 − t)(1 − t^{4})(1 − t^{6}).}

Proof: As already stated, this follows from the double coset formula. Note that s_{1}^{3} = s_{1}^{2}t_{1}+s_{1}t_{2} and the terms on the right are already represented. Simliarly s_{1}L = s_{1}^{2}s_{2}+s_{1}t_{1}s_{2} and these terms are represented. Thus the second summand in the given expression is all multiples of s_{1} in H^{∗}(H; /2). The group M_{12}

From 4.10, 4.14 we have the image of H^{∗}(W; /2), H^{∗}(W′; /2) in H^{∗}(H; /2). But the intersection of these two images is H^{∗}(M_{12}; /2). We now list the generators in H^{∗}(M_{12}; /2) through dimension 7:

4.15

Name  Generator  Dimension
α_{2} s_{2} + s_{1}^{2}  2
x_{3} s_{1}s_{2}  3
y_{3} (s_{1}^{2} + s_{2})t_{1} + s_{1}t_{2}  3
z_{3} L + (s_{1} + t_{1} + e_{1})t_{2} + s_{1}t_{1}^{2}  3
β_{4} k_{4} + t_{2}^{2} + e_{1}^{4} + t_{1}^{4}  4
γ_{5}  {(t_{2} + e_{1}^{2} + t_{1}^{2})L + e_{1}(k_{4} + t_{2}^{2} + e_{1}^{2}t_{2}) \atop +t_{1}t_{2}^{2} + t_{1}^{3}t_{2} + s_{1}t_{2}^{2} + s_{1}^{2}t_{1}t_{2} + t_{1}s_{2}t_{2} + s_{1}t_{1}^{4}}  5
δ_{6} (t_{2} + e_{1}^{2} + t_{1}^{2})k_{4} + (e_{1}^{2} + t_{1}^{2})t_{2}^{2}  6
ϵ_{7} (t_{1} + e_{1})t_{2}k_{4}  7

Note that since M_{12} is simple H^{1}(M_{12}; /2) = 0. We also have the Steenrod operations

Sq^{1}(α) = x Sq^{2}(α) = α^{2}
Sq^{1}(x) = 0 Sq^{2}(x) = αx  Sq^{3}(x) = x^{2}
Sq^{1}(y) = α^{2} Sq^{2}(y) = αz = α(x + y) Sq^{3}(y) = y^{2}
Sq^{1}(z) = α^{2} Sq^{2}(z) = γ  Sq^{3}(z) = z^{2}
Sq^{1}(β) = 0 Sq^{2}(β) = δ  Sq^{3}(β) = ϵ Sq^{4}(β) = β^{2}
Sq^{1}(γ) = z^{2} Sq^{2}(γ) = 0  Sq^{3}(γ) = 0 Sq^{4}(γ) = βγ + zδ + αϵ
Sq^{1}(δ) = ϵ Sq^{2}(δ) = 0  Sq^{3}(δ) = 0 Sq^{4}(δ) = βδ
Sq^{1}(ϵ) = 0 Sq^{2}(ϵ) = 0  Sq^{3}(ϵ) = 0 Sq^{4}(ϵ) = βϵ,

and the following are a complete set of relations among the generators in 4.15

\begin{matrix}α(x + y + z) = 0 & x^{3} = α^{3}x + αβx + xδ \\ xy = α^{3} + x^{2} + y^{2} & xz = α^{3} + y^{2} \\ x^{2}y = α^{3}z + αβz + yδ + αϵ & yz = α^{3} + x^{2} \\ ϵx = βx^{2} + α^{2}x^{2} & αγ = α^{2}y \\ ϵy = α^{2}δ + α^{2}y^{2} + βx^{2} + βy^{2} & yγ = αy^{2} \\ ϵz = γ^{2} + α^{2}δ + α^{2}x^{2} + βx^{2} + βz^{2} & xγ = α^{4} + αz^{2} \\ z^{4} = γϵ + x^{4} + α^{4}β + z^{2}δ & \begin{matrix}ϵ^{2} = & z^{3}γ + α^{2}βδ + α^{5}β + zβϵ + zδ(γ + αz) \\ + β^{2}(α^{3} + xz + yz). \end{matrix} \end{matrix}

This result was checked by the program MACAULAY, and we thank David Rusin for his help at this point. The generators in 4.15 above were analyzed by the program and shown to satisfy exactly this set of relations in H^{∗}(H; /2). Moreover, the program calculated the Poincaré series for this algebra as

4.16 q(t) = \frac{1 + t^{2} + 3t^{3} + t^{4} + 3t^{5} + 4t^{6} + 2t^{7} + 4t^{8} + 3t^{9} + t^{10} + 3t^{11} + t^{12} + t^{14}}{(1 − t^{4})(1 − t^{6})(1 − t^{7})}

Then the program was used to show Theorem 4.17: The algebra above is Cohen-Macaulay. Specifically, it is free and finitely generated as a module over the polynomial algebra _{2}[β_{4}, δ_{6}, ϵ_{7}].

Using a result due to Webb [We], we now derive the Poincaré series for H^{∗}(M_{12}; /2). Write G = M_{12}, X = |A_{2}(M_{12})| with the conjugation action. Then Webb’s result is

Ĥ^{∗}(G; /2) ⊕ ⨁_{\begin{matrix}σ_{i} i−cell \\ in X/G \\ i odd \end{matrix}} Ĥ^{∗}(G_{σ_{i}}; /2) ≅ ⨁_{\begin{matrix}σ_{i} i−cell \\ in X/G \\ i even \end{matrix}} Ĥ^{∗}(G_{σ_{i}}; /2).

Using our explicit knowledge of the equivariant structure of X, we obtain many cancellations, effectively simplifying the formula to yield

H^{∗}(M_{12}) ⊕ H^{∗}(H) ≅ H^{∗}(W) ⊕ H^{∗}(W′).

This involves cancelling isotropy groups for vertices with edges, and other isotropy groups of edges with isotropy groups of 2-simplexes. After that a few explicit calculations are required to show that the mod 2 cohomology rings of the isotropy groups of most of the remaining edges cancel off with those for remaining vertices and 2-simplices.

From our previous calculations this can be used to determine the Poincaré series for M_{12}. After some simplification we obtain Corollary 4.18: The Poincaré series for H^{∗}(M_{12}; /2) is

p(t) = \frac{1 + 3t^{3} + 3t^{6} + t^{9}}{(1 − t^{2})(1 − t^{4})(1 − t^{6})} = \frac{(1 + t^{3})^{2}}{(1 − t^{2})(1 − t^{3})(1 − t^{4})}.

Then we compare the polynomials in 4.18 and 4.16. It is direct to see that they are identical, and from this the proof that H^{∗}(M_{12}; /2) is the algebra above is complete.

It would be useful to understand, in some sense, meanings for the generating classes and their relations, for example, in terms of characteristic classes for representations or images of classes from H^{∗}(S_{12}; /2), a topic we hope to consider in a sequel.

§5. TRIVALENT GRAPHS AND THE AMALGAMATED PRODUCT

Given a situation such as that of H, W, and W′, we can find a universal completion Γ = W ∗_{H} W′ which makes the diagram below commute,

H ↪ W′
↓  ↓ϕ_{2}

W ⟶^{ϕ_{1}}  Γ

and such that any Γ′ which satisfies this (generated by W and W′) is a quotient of Γ. Γ is called the amalgamated product of W and W′ over H. It is well known, (see [Se]), that an amalgamated product as above will act on a tree with finite isotropy, and orbit space of the form

W  ^{H}  W′

In [G], Goldschmidt analyzed the situation for actions on the cubic tree (the tree of valence 3) and obtained a classification of finite primitive amalgams of index (3, 3) (this refers to the indexes [W : H], [W′ : H]). He shows that M_{12} is one of 15 such amalgams, necessarily a quotient of the universal one Γ.

From this we deduce the existence of an extension

5.1  1⟶Γ′⟶Γ⟶M_{12}⟶1

where Γ′ is a free group (it has cohomological dimension 1). Using the formula for Euler characteristics in [B], we have, on the one hand

χ(Γ) = \frac{1}{192} + \frac{1}{192} − \frac{1}{64} = −\frac{1}{192}

(amalgamated product), and also

χ(Γ) = \frac{χ(Γ′)}{|M_{12}|}.

Hence χ(Γ′) = 95, 040(−\frac{1}{192}) = −495 and it follows that Γ′ ≅ ∗_{1}^{496} , the free group on 496 generators.

We can now state the main result of this section Theorem 5.1: The map Γ⟶M_{12} induces an isomorphism

H^{∗}(M_{12}; /2)⟶H^{∗}(Γ; /2).

Proof: Consider the map

res_{H}^{W} ⊕ res_{H}^{W′} : H^{∗}(W) ⊕ H^{∗}(W′)⟶H^{∗}(H).

Its kernel is clearly im(res_{H}^{W}) ∩ im(res_{H}^{W′}) ≅ H^{∗}(M_{12}) by 3.1. On the other hand, from [We] we have seen that H^{∗}(W) ⊕ H^{∗}(W′) ≅ H^{∗}(M_{12}) ⊕ H^{∗}(H). Hence res_{H}^{W} ⊕ res_{H}^{W′} is onto. On the other hand, from the action of Γ on a tree described previously, there is a well known long exact sequence

⋯ ⟶H^{i}(Γ)⟶H^{i}(W) ⊕ H^{i}(W′)⟶H^{i}(H)⟶H^{i+1}(Γ)⟶ ⋯

As it comes from a Mayer-Vietoris sequence the same map as before arises, hence the sequence splits and

H^{∗}(W) ⊕ H^{∗}(W′) ≅ H^{∗}(Γ) ⊕ H^{∗}(H).

Consequently, by rank considerations and the fact that the finite subgroups in Γ are mapped isomorphically into M_{12} under the projections the proof is complete. Corollary 5.3: H^{1}(K; /2) is an M_{12}-acyclic /2(M_{12})-module of rank 496 which is not projective. Proof: The proof follows from considering the spectral sequence over /2 associated to (5.1) and the observation that 64 does not divide 496.

We do not know whether or not this representation has been documented in the literature, but it has radically different cohomological behavior at distinct primes dividing |M_{12}|. For example, at p = 3 we have a sequence

H^{p−2}(M_{12}; H^{1}(K; /3))⟶H^{p}(M_{12}; /3)⟶H^{p}(W; /3) ⊕ H^{p}(W′; /3)

and clearly the term on the left must be non-trivial.

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Mathematics Department Stanford University Stanford, CA 94305 Mathematics Department University of Michigan Ann Arbor, Mich. 48109-1003 Mathematics Department Stanford University Stanford, CA 94305