The 3-local cohomology of the Mathieu group M_{24}

David John Green

Institut für Experimentelle Mathematik

Universität GHS Essen Ellernstraße 29 D–45326 Essen

Germany

Email: david@exp-math.uni-essen.de

11^{th} November 1994

To appear in Glasgow Math. J.

Introduction. In this paper we calculate the localisation at the prime 3 of the integral cohomology ring of the Mathieu group M_{24}, together with its mod-3 cohomology ring. The main results are:

Theorem 1. The ring H^{∗}(M_{24}, Z)_{(3)} is the commutative graded Z_{(3)}- algebra with generators

Generator  β  θ  ν  ξ
Degree  4 16 11 12
Additive order 3  3  3  3^{2}

and relations ν^{2} = 0 and βθ = 0. The Chern classes of the Todd representation in GL_{11} F_{2} generate the even-degree part of this ring.

Theorem 2. The commutative graded F_{3}-algebra H^{∗}(M_{24}, F_{3}) has generators

Generator B b N n, X  x  T  t
Degree  3 4 10  11  12 15 16

and relations

Bn = bN  T X = T n = tN  tX = tn

n^{2} = B^{2} = T^{2} = N^{2} = bt = bT = nN = tB = BN = BT = NT = 0

bX = nX = BX = NX = X^{2} = 0.

In [9], Thomas uses our results to prove that the elliptic cohomology of the classifying space BM_{24} is generated by Chern classes, and is therefore concentrated in even dimensions.

1. The Mathieu group M_{24}. The Mathieu group M_{24} is a 5-transitive degree 24 permutation group of order 2^{10}.3^{3}.5.7.11.23. We can read off the 3-local structure we require from the Atlas [2]. The Sylow 3-subgroups are isomorphic to 3_{+}^{1+2}, the extraspecial 3-group of order 3^{3} and exponent 3. This has a presentation

3_{+}^{1+2} ≅ ⟨A, B, C | A^{3} = B^{3} = C^{3} = 1, CA = AC, CB = BC, AB = BAC⟩.

Let P be a Sylow 3-subgroup of G. We see that each 3^{2} is self-centralising, and that the Sylow 3-normaliser N = N_{G}(P) is isomorphic to 3_{+}^{1+2} : D_{8}. The outer automorphism group of 3_{+}^{1+2} is isomorphic to GL_{2}F_{3}, which has Sylow 2-subgroups isomorphic to the semidihedral group SD_{16}. As SD_{16} has exactly one subgroup isomorphic to D_{8}, there is only one conjugacy class of subgroups of GL_{2}F_{3} isomorphic to D_{8}. Hence, choosing new generators for P if necessary, we may assume that the D_{8} is generated by elements J and K as follows: conjugation by J sends A to B^{2}, sends B to A and fixes C; and conjugation by K sends A to B^{2}, sends B to A^{2} and C to C^{2}.

There are two conjugacy classes of elements of order 3 in M_{24}. We may assume that we have chosen generators for P and N/P such that in P, the elements of class 3A are C^{r}, A^{r}C^{t} and B^{r}C^{t}, whereas A^{r}B^{r}C^{t} and A^{r}B^{−r}C^{t} have class 3B. Here r ∈ {1, 2} and t ∈ {0, 1, 2}.

2. The 3-local integral cohomology. We shall now calculate the 3-local integral cohomology ring, using a well-known result from the book of Cartan and Eilenberg.

Theorem 3. ([1]) Let G be a finite group with Sylow p-subgroup P. Recall that a class x in H^{∗}(P, Z)_{(p)} is stable if, for each g in G, the image (under conjugation by g) of x in H^{∗}(P^{g}, Z)_{(p)} has the same restriction to P ∩ P^{g} as has x itself.

The restriction map from G to P is an isomorphism between H^{∗}(G, Z)_{(p)} and the ring of stable classes in H^{∗}(P, Z)_{(p)}.

Here P is 3_{+}^{1+2}, whose integral cohomology was calculated by Lewis.

Theorem 4. ([6]) The cohomology ring H^{∗}(3_{+}^{1+2}, Z) is generated by

Generator  α_{1}, α_{2} ν_{1}, ν_{2} κ ζ
Degree  2  3  4  6
Additive order 3  3  3 3^{2}

.

The ν_{i} square to zero. The remaining relations are:

\begin{matrix}α_{i}κ = & −α_{i}^{3} & α_{1}ν_{2} = α_{2}ν_{1} & α_{1}α_{2}^{3} = & α_{1}^{3}α_{2} \\ ν_{i}κ = & −α_{i}^{2}ν_{i} & κ^{2} = α_{1}^{4} − α_{1}^{2}α_{2}^{2} + α_{2}^{4} & α_{2}^{3}ν_{1} = & α_{1}^{3}ν_{2} \end{matrix}  ν_{1}ν_{2} = ±3ζ.

The automorphism which sends A to A^{r′}B^{s′}C^{t′}, B to A^{r}B^{s}C^{t} and C to C^{j} fixes κ, sends ζ to j^{3}ζ and sends

\begin{matrix}α_{1} & ↦ r′α_{1} + rα_{2} \\ α_{2} & ↦ s′α_{1} + sα_{2} \end{matrix}  \begin{matrix}ν_{1} & ↦ j(r′ν_{1} + rν_{2}) \\ ν_{2} & ↦ j(s′ν_{1} + sν_{2}) . \end{matrix}

We start by calculating the cohomology of N: this is the ring of classes in H^{∗}(P, Z)_{(3)} which are invariant under the action of the Sylow 3-normaliser, i.e., under conjugation by J and K.

Proposition 5. The ring H^{∗}(N, Z)_{(3)} is generated by α = α_{1}^{2} + α_{2}^{2}, κ, η = ζ^{2} and ν = (α_{1}ν_{1}+α_{2}ν_{2})ζ. Additive exponents are obvious, and ν squares to zero. The other relation is α^{2} = κ^{2}.

Proof. We wish to diagonalise the action of J. Write H_{3} for the module generated by the α_{j} and the ν_{j} over the ring generated by the α_{j}. Then H_{3} is an F_{3}-vector space, and additively a direct summand of H^{∗}(P, Z)_{(3)}. Extending the scalars to F_{9} makes the action of J diagonalisable. Write i for a primitive fourth root of unity in F_{9}.

J fixes κ and ζ, multiplies α_{1} − iα_{2} by i, and α_{2} + iα_{2} by −i. Hence in even degree the fixed classes are generated by κ, ζ, α_{1}^{2} + α_{2}^{2} and (α_{1} ∓ iα_{2})^{4}. In both cases this last expression is α_{1}^{4} + α_{2}^{4}, which is −κα. Similarily, the only odd-degree generator needed is α_{1}ν_{1} + α_{2}ν_{2}, which we call μ.

K fixes κ and α, and multiplies ζ and μ by −1, whence the result.

We now obtain a lower bound for the even-degree cohomology of G: in fact this bound is attained.

Proposition 6. The Chern subring of G contains β = α+κ, ξ = η −κ^{3} and θ = (α − κ)η.

Proof. Consider the Todd representation of G in GL_{11} F_{2}. After lifting to characteristic zero (see [8], [4]), we obtain a generalised character χ_{τ} with partial character table

\begin{aligned}& 1A & 3A & 3B \\ χ_{τ} & 11 & 2 & −1 \end{aligned} .

The irreducible representations of 3_{+}^{1+2} are ρ^{xy} for 0 ≤ x, y ≤ 2, and ρ^{z} for 1 ≤ z ≤ 2. They have characters

χ^{xy}: A^{r}B^{s}C^{t} ⟼ ω^{rx+sy}
χ^{z}: A^{r}B^{s}C^{t} ⟼ \{ \begin{matrix}3ω^{zt} & r = s = 0 \\ 0 & otherwise, \end{matrix}

where ω is of course exp{2πi/3}. We have χ_{τ} = χ^{00} + χ^{10} + χ^{20} + χ^{01} + χ^{02} + χ^{1} + χ^{2}. Let ρ_{τ} be a virtual representation affording χ_{τ}.

Theorem 7. ([5]) These irreducible representations of 3_{+}^{1+2} have total Chern classes c(ρ^{xy}) = 1 + xα_{1} + yα_{2} and c(ρ^{z}) = 1 + κ + z^{3}ζ.

Using the Whitney sum formula,

c(ρ_{τ}) = (1 − α_{1}^{2})(1 − α_{2}^{2})(1 − κ + κ^{2} − ζ^{2})
= 1 − (α + κ) − (η − κ^{3}) + (αη − ακ^{3} − κ^{4}) + (ακ + κ^{2})η.

So c_{2}(ρ_{τ}) = −β, c_{6} = −ξ, and c_{8} = −(θ + βξ + β^{4}).

In general, H^{∗}(N, Z)_{(3)} need not be closed under the action on H^{∗}(P, Z)_{(3)} of an automorphism of P. However, in the proof of Theorem 1 we will need to be able to approximate any automorphism of P by one that does act on H^{∗}(N, Z)_{(3)}.

Lemma 8. Let ϕ be an automorphism of P, and D a non-central element of P. Then there is an automorphism ψ of P such that ψ equals ϕ on ⟨D, C⟩, and also H^{∗}(N, Z)_{(3)} is closed under the action of ψ on H^{∗}(P, Z)_{(3)}. The map ψ^{∗} fixes κ and η, and multiplies α and ν by ϵ, where ϵ is +1 or −1 according as D and ϕD are in the same or different conjugacy classes of G.

Proof. The automorphism group of P acts transitively on the non-central elements, and hence transitively on the subgroups of order 3^{2}. Therefore it suffices to prove the lemma for D = B.

Let ϕB be A^{r}B^{s}C^{t}, and let ϕC be C^{j}. We shall find a and b such that defining ψA to be A^{a}B^{b} gives us an automorphism ψ with the required properties. For ψ to be well-defined, we need j ≡ as − rb. Now, ψ^{∗} sends α = α_{1}^{2} + α_{2}^{2} to (a^{2} + b^{2})α_{1}^{2} − (ar + bs)α_{1}α_{2} + (r^{2} + s^{2})α_{2}^{2}. There is a unique solution modulo 3 to the equations as − rb ≡ j and ar + bs ≡ 0. This also satisfies a^{2} + b^{2} = r^{2} + s^{2}. Hence ψ^{∗}α is in H^{∗}(N, Z)_{(3)}, and ψ^{∗}ν is too. Finally, κ and η are fixed by all automorphisms of P, and r^{2} + s^{2} is +1 or −1 according as ϕB is in 3A or 3B.

Proposition 9. ([6]) Let D be A^{r}B^{s}C^{t}. The ring H^{∗}(C_{3}^{D} × C_{3}^{C}, Z)_{(3)} is generated by δ and γ in degree 2, and χ in degree 3. All three generators have additive order 3, and χ squares to zero. The automorphism of C_{3} × C_{3} which switches the two factors sends δ ↔ γ and χ ↦ −χ. Restriction from P sends α_{1} to rδ, ν_{1} to rχ, α_{2} to sδ, ν_{2} to sχ, κ to −δ^{2} and ζ to γ^{3} − γδ^{2}.

Proof of Theorem 1. We have to obtain the stable classes in H^{∗}(P, Z)_{(3)}. In Proposition 5 we calculated H^{∗}(N, Z)_{(3)}, which consists of those classes which are stable with respect to each g in N_{G}(P). We now have to consider each g which is not in N_{G}(P). We can ignore those P^{g} whose intersection with P has order 3, because corestriction from C_{3} to C_{3} × C_{3} is zero. (See Proposition 18 of [3].)

So we may suppose that P^{g} ∩ P has order 3^{2}. Such g do exist, because G contains 3^{2} : GL_{2}F_{3}. The groups 3^{2} in G contain either two or eight elements of class 3A, and the centre of a Sylow 3-subgroup contains two elements of 3A. Now P and P^{g} cannot have the same centre, for both would have to lie in the centraliser in G of a 3A: this is the triple cover ˆ3.A_{6}, but A_{6} is T.I. at 3. Hence P ∩ P^{g} contains eight elements of class 3A, and is therefore ⟨A, C⟩ or ⟨B, C⟩.

Suppose that P ∩ P^{g} is ⟨D, C⟩, with D = A^{r}B^{s}C^{t} central in P^{g}. So D is in 3A. Lemma 8 allows us to construct an automorphism ψ of P with ψD = gCg^{−1} and ψC = gDg^{−1}, such that ψ^{∗} fixes every element of H^{∗}(N, Z)_{(3)}. Let f be the automorphism of ⟨D, C⟩ which switches the two factors around.

Including ⟨D, C⟩ in P^{g} and then conjugating by g is the same map to P as applying f, then including in P and then applying ψ. So a class x in H^{∗}(N, Z)_{(3)} is in H^{∗}(G, Z)_{(3)} if and only if its restriction to ⟨D, C⟩ is fixed by f^{∗}.

Since r^{2} + s^{2} ≡ 1 (mod 3), restriction sends α to δ^{2}, ν to δγ(γ^{2} − δ^{2})χ, κ to −δ^{2} and η to γ^{2}(γ^{2} − δ^{2})^{2}. We immediately see that ν is stable, and generates the odd-degree stable classes over the even-degree stable classes. We know from Proposition 6 that α + κ, η − κ^{3} and (α − κ)η are stable, and we can now easily verify this. We claim that these three classes generated the even-degree stable classes. Since (α + κ)(η − κ^{3}) − (α + κ)^{4} = (α + κ)η, they certainly generate κη.

Let x be a (homogeneous) stable class of even degree. Subtracting powers of η − κ^{3} if necessary, x contains no lone powers of η (i.e., x involves no momonial of the form η^{ℓ}). Since (α + κ)^{t+1} = (−1)^{t}κ^{t}(α + κ), we may further assume that x contains no lone powers of κ. Then x cannot contain a lone ακ^{t}, because the restriction of x would contain a lone power of δ without the corresponding power of γ required for being fixed by f^{∗}. Hence every term in x is divisible by αη or κη. Since α^{2} = κ^{2}, the only terms not divisible by κη are of the form αη^{t+1}, which can be eradicated by subtracting (α − κ)η(η − κ^{3})^{t}. So x can be reduced to κηx′. Then x′ is stable, and x is a polynomial in our supposed generators if x′ is. Since x′ has lower degree, the claim follows by induction. Finally, the relations are obvious.

3. The mod-3 cohomology. Recall that to the short exact sequence

0 → Z_{(3)} →^{3×} Z_{(3)} →^{j} F_{3} → 0  (1)

of coefficient modules there is an associated long exact sequence

⋯ →^{∂} H^{n}(G, Z)_{(3)} →^{3×} H^{n}(G, Z)_{(3)} →^{j_{∗}} H^{n}(G, F_{3}) →^{∂} H^{n+1}(G, Z)_{(3)} →^{3×} ⋯ (2)

of cohomology groups. Using the properties of this long exact sequence, we shall derive the structure of H^{∗}(M_{24}, F_{3}) from that of H^{∗}(M_{24}, Z)_{(3)}.

Recall that the Bockstein homomorphism Δ = j_{∗} ∘ ∂ is a graded derivation, and that the connecting map ∂ has a property akin to Frobenius reciprocity: if x ∈ H^{n}(G, F_{3}) and y ∈ H^{m}(G, Z)_{(3)}, then ∂(xj_{∗}(y)) = ∂(x)y.

First we derive the Poincaré series of H^{∗}(M_{24}, F_{3}):

Theorem 10. The F_{3}-cohomology ring of M_{24} has Poincaré series

\frac{1 + t^{3} + t^{4} + t^{7} + t^{8} + t^{10} + 3t^{11} + t^{12} + t^{14} + 3t^{15} + t^{16} + t^{18} + t^{19} + t^{22} + t^{23} + t^{26}}{(1 − t^{12})(1 − t^{16})} .

Proof. Consider the long exact sequence (2) of cohomology groups. Each non-zero monomial in the generators of Theorem 1, lying in H^{n}(G, Z)_{(3)}, contributes one basis vector to H^{n−1}(G, F_{3}), and one to H^{n}(G, F_{3}). This does apply to the ξ^{ℓ}, but naturally not to 1. So we calculate the generating function f(t) for the number of non-zero monomials which lie in H^{n}(G, Z)_{(3)}.

If the only generator were β, then f(t) would be 1/(1 − t^{4}); if θ were the only generator, it would be 1/(1−t^{16}). Since βθ = 0, the generating function for the subring they together generate is

\frac{1}{1 − t^{4}} − 1 + \frac{1}{1 − t^{16}} = \frac{1 + t^{4} + t^{8} + t^{12} + t^{16}}{1 − t^{16}} .

The subrings generated by ν and by ξ have generating functions 1 + t^{11} and 1/(1 − t^{12}) respectively. Since we have already budgeted for all the relations, we have

f(t) = \frac{1 + t^{4} + t^{8} + t^{12} + t^{16}}{1 − t^{16}} × (1 + t^{11})× = \frac{1}{1 − t^{12}} .

By the argument at the start of this proof, the desired Poincaré series is then f(t) + (f(t) − 1)/t.

Proof of Theorem 2. We use the cohomology long exact sequence (2) associated to the short exact sequence (1) of coefficient modules. Define b = j_{∗}(β), t = j_{∗}(θ), n = j_{∗}(ν) and x = j_{∗}(ξ). By exactness there are unique B ∈ H^{3}(G, F_{3}) and N ∈ H^{10} such that ∂(B) = β and ∂(N) = ν. We want T such that ∂(T) = θ. This only defines T up to adding a multiple of bn. Since BT is in H^{18}, which has basis bnB, there is a unique T in H^{15} satisfying ∂(T) = θ and BT = 0. There is similarily a unique X̄ in H^{11} defined by ∂(X̄) = 3ξ and BX̄ = 0. We shall set X = ±X̄, with the sign to be determined later. Since the image of ∂ is the ideal in H^{∗}(G, Z)_{(3)} generated by β, θ, ν and 3ξ, we have a complete set of generators.

Most relations follow immediately. To prove that bT is zero, note that it lies in H^{19}, which is an F_{3}-vector space with basis b^{2}n, b^{4}B, bxB. Applying ∂ demonstrates that bT is a scalar multiple of b^{2}n. Multiplying by B then shows that bT is zero, for ∂(Bb^{2}n) = β^{3}ν, which is non-zero.

Since N^{2} lies in H^{20}, it is a linear combination of b^{5} and b^{2}x. Multiplication by n shows that N^{2} is zero, since nN is zero for degree reasons.

To prove that T X = T n and tX = tn, we need a more intricate argument. Since T X lies in H^{26}, it must be an F_{3}-linear combination of Bb^{3}n, Bxn and T n. Since bT = 0, multiplication by b shows that T X must be a scalar multiple of T n. Applying the Bockstein map, tX is the same multiple of tn. Since we can choose X = ±X̄, it is enough to prove that tX̄ ≠ 0.

Let D = A^{r}B^{s}C^{t} be a non-central element of P. When we restrict from G to ⟨D, C⟩, by Proposition 9 we have

Res β = (r^{2} + s^{2} − 1)δ^{2}  Res θ = (r^{2} + s^{2} + 1)γ^{2}δ^{2}(γ^{2} − δ^{2})^{2}. (3)

Observe that r^{2} + s^{2} ≡ 1 (mod 3) if D is of class 3A, and −1 if D is 3B. We have

t = \{ \begin{matrix}−j_{∗}(γ^{2}δ^{2}(γ^{2} − δ^{2})^{2}) & D ∈ 3A \\ 0 & D ∈ 3B \end{matrix} .

Hence if D ∈ 3A, Res t is neither zero nor a zero divisor. For if y ∈ H^{m}(⟨D, C⟩, F_{3}) with m > 0 and ty = 0, then ∂(ty) = 0, and so γ^{2}δ^{2}(γ^{2} − δ^{2})^{2} ∂(y) = 0. If follows quickly from Proposition 9 that ∂(y) = 0, and so y = j_{∗}(υ) for some υ ∈ H^{m}(⟨D, C⟩, Z)_{(3)}. Since j_{∗} is an injection here, it follows from ty = 0 that γ^{2}δ^{2}(γ^{2} − δ^{2})^{2}υ = 0, whence υ = 0 and y = 0. So it is enough to prove that, for some D ∈ 3A, Res X̄ ≠ 0.

Similarily, b = \{ \begin{matrix}0 & D ∈ 3A \\ j_{∗}(δ^{2}) & D ∈ 3B \end{matrix} . So, if D ∈ 3B, then, as above, Res b

is neither zero nor a zero divisor. But bX̄ = 0, and so if D ∈ 3B then Res X̄ = 0.

A result of Milgram and Tezuka [7] states that the maximal elementary abelian subgroups of 3_{+}^{1+2} detect every non-zero element of H^{∗}(3_{+}^{1+2}, F_{3}). Hence, for some D ∈ 3A, Res X̄ ≠ 0. Note that in the special case of X̄, Milgram and Tezuka’s result can be quickly verified. For X̄ ∈ H^{11}(P, F_{3}) is non-zero and in the kernel of Δ, and so is a non-zero F_{3}-linear combination of the images under j_{∗} of α_{1}^{4}ν_{1}, α_{1}^{3}α_{2}ν_{1}, α_{1}^{2}α_{2}^{2}ν_{1}, α_{2}^{4}ν_{2}, α_{1}ν_{1}ζ, α_{2}ν_{1}ζ and α_{2}ν_{2}ζ. But we can quickly check from Proposition 9 that any non-zero F_{3}-linear combination of these elements is detected by restriction to the four maximal elementary abelian subgroups.

We have now established the claimed relations. These show us that, as a module over the ring generated by x and b^{4} + t, H^{∗}(G, F_{3}) is generated as a module by the twenty elements 1, B, b, bB, b^{2}, b^{2}B, b^{3}, b^{3}B, T, t, N, n, nB, bn, bnB, b^{2}n, b^{2}nB, b^{3}n, b^{3}nB, X. As the free module with these generators has the correct Poincaré series, there are no further relations.

Remark. The author is grateful to the referee for the observation that the ring H^{∗}(Aut(M_{12}), F_{3}) is isomorphic to H^{∗}(M_{24}, F_{3}). For we see from the Atlas [2] that M_{24} has a maximal subgroup isomorphic to Aut(M_{12}), and that this contains copies of both 3_{+}^{1+2} : D_{8} and 3^{2} : GL_{2}F_{3}. Consequently, we may apply the proof of Theorem 1 to Aut(M_{12}) and deduce that restriction from H^{∗}(M_{24}, Z)_{(3)} to H^{∗}(Aut(M_{12}), Z)_{(3)} is a ring isomorphism. Now, the only information about M_{24} that we use in calculating H^{∗}(M_{24}, F_{3}) is the structure of the ring H^{∗}(M_{24}, Z)_{(3)} and the fact that the Sylow 3-subgroups of M_{24} are isomorphic to 3_{+}^{1+2}. It therefore follows that H^{∗}(Aut(M_{12}), F_{3}) is isomorphic to H^{∗}(M_{24}, F_{3}).

Acknowledgements. The author would like to thank W. Lempken for explaining the 3-local structure of M_{24}, and the referee for the above remark. The author is grateful for the support of the Deutsche Forschungsgemeinschaft.

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