The Cohomology of the Mathieu Group M_{23}

R. James Milgram†

M_{23} is the Mathieu group of order 2^{7} ·3^{2} ·5·7·11·23, one of the 26 sporadic groups. In this note we determine the cohomology rings H^{∗}(M_{23}; _{p}) for all primes p dividing |M_{23}|. This has several interesting consequences.

M_{23} is somewhat unusual among the sporadic groups in that

Out(M_{23}) = Mult(M_{23}) = 1,

and, as a result of our calculation H_{i}(M_{23}; ) = 0 for i < 5. In particular M_{23} is the first known counterexample to the conjecture (C. Giffin, [G]), that if G is a finite group with H_{i}(G; ) = 0, i = 1, 2, 3, then G = {1}. (M_{11}, the first Mathieu group and J_{1}, the first Janko group also satisfy Out(G) = Mult(G) = 1, but for both of these groups H_{3}(G; ) ≠ 0.)

At the prime two we have that H_{6}(M_{23}; ) = /2 is the first non-zero homology group. In particular, when we look at the usual inclusion M_{23} ⊂ S_{23} we can ask about the image of this first non-trivial class. We have the well known equivalence B_{S_{∞}}^{+} ≃ Q_{0}(S^{0}), the 0-component of the infinite loop space lim_{n↦∞}(Ω^{n}S^{n}), and at the prime two, Q_{0}(S^{0})_{2} ≃ im(J)_{2} × coker(J)_{2} is Quillen’s splitting, [Q], [BM]. Then coker(J)_{2} is 5-connected and we have

Theorem: The composition

B_{M_{23}}^{+}⟶B_{S_{23}}^{+}⟶B_{S_{∞}}^{+} = Q_{0}S^{0}

induces an injection in dimension six from π_{6}(B_{M_{23}}^{+})⟶π_{6}(coker(J)) = /2. In particular one of the roles of M_{23} is to construct the first non-trivial element in π_{∗}(coker(J)).

Remark: One might conjecture that the sporadic groups and the exceptional finite Chevellay groups should play a basic role (through the +-construction and various homomorphisms of the groups in question into symmetric groups) in the structure of the space coker(J). Indeed, the author has conjectured exactly this, in part based on Quillen’s result that the groups O_{n}( _{3}), and GL_{n}( _{p}) determine im(J) in a similar way. But this result for M_{23} is the most interesting partial confirmation of that guess so far.

Remark: I have not computed the full map from H^{∗}(S_{23}; _{2}) to H^{∗}(M_{23}; _{2}) yet, but, in view of the calculations here it should be quite direct to obtain the entire answer. It cannot be the case that H^{∗}(coker(J); _{2})⟶H^{∗}(M_{23}; _{2}) is onto, but it appears that it should be onto except for a very interesting piece which might be closely tied in to the Dwyer-Wilkerson space, [DW].

†Partially supported by a grant from the N.S.F

In §4, we study the map H^{∗}(S_{23}; _{2})⟶H^{∗}(M_{23}; _{2}). Then, in §5 we also apply our techniques to determine the map H^{∗}(S_{12}; _{2})⟶H^{∗}(M_{12}; _{2}). Here the connection with the homotopy groups of spheres is less direct, but it seems likely that the composition

M_{12} ↪ M_{24} ↪ S_{24}

ties B_{M_{12}}^{+} with a particular element in π_{14}^{s}(S^{0}), the element σ^{2}.

§1. The main results

Our analysis breaks naturally into two parts, the calculation at the prime 2, and the calculation at odd primes. The results at 2 are heavily dependent on the analysis in [AM3] for H^{∗}(M_{22}; _{2}).

The results at 2

Recall that H^{∗}(2^{n}; _{2})^{GL_{n}(2)} = _{2}[d_{2^{n−1}}, d_{2^{n−1}+2^{n−2}}, …, d_{2^{n}−1}], the Dickson algebra, where the Dickson generators are connected by the operation of the Steenrod squares Sq^{m−i}(d_{i}) = d_{m}, and d_{2^{n−1}} is the symmetric sum ∑_{i_{1}≥···≥i_{n}} Sx_{1}^{i_{1}} ⋯ x_{n}^{i_{n}} where each i_{j} is a power of 2. For example d_{2} = Sx_{1}^{2} + x_{1}x_{2} = x_{1}^{2} + x_{2}^{2} + x_{1}x_{2}, d_{4} = Sx_{1}^{4} + Sx_{1}^{2}x_{2}^{2} + Sx_{1}^{2}x_{2}x_{3} + x_{1}x_{2}x_{3}x_{4}.

In [AM3] the cohomology of M_{22} is determined as follows. First there are three conjugacy classes of extremal elementary 2 groups in M_{22}, given as E, F, both ≅ 2^{4}, and V ≅ 2^{3}. Their normalizers are respectively E: A_{6}, F: S_{5}, and V: GL_{3}(2). The image of the sum of the three restrictions

\overline{res}^{∗}: H^{∗}(M_{22}; _{2})⟶H^{∗}(E)^{A_{6}} ⊕ H^{∗}(F)^{S_{5}} ⊕ H^{∗}(V; _{2})^{GL_{3}(2)}

is given explicitly in [AM3], aided considerably by the fact that all three are weakly closed as subgroups of Syl_{2}(M_{22}) ⊂ M_{22}‡ while Ker(\overline{res}^{∗}) injects into the cohomology ring H^{∗}(Syl_{2}(L_{3}(4)); _{2}) and has the form

2[d̄_{8}, d̄_{12}](a_{2}, a_{7}, a_{11}, a_{14}).

In M_{23} these three subgroups remain extremal. This is clear for E and F since they are not conjugate in M_{23} from [Co], and Syl_{2}(M_{22}) = Syl_{2}(M_{23}) only has two 2^{4} subgroups from [J]. For V the argument is given in (3.6). But in M_{23} the Weyl group of E, W_{M_{23}}(E) = N_{M_{23}}(E)/E is A_{7} and we show in (2.1) that the Weyl group of F is GL_{2}( _{4}): 2 where the action of the element of order 2 is by Galois conjugation on the entries of the matrices in GL_{2}( _{4}). Moreover, each of these three extremal groups is weakly closed in Syl_{2}(M_{23}) ⊂ M_{23}, consequently, we obtain the following description of H^{∗}(M_{23}; _{2}).

‡A subgroup V ⊂ L ⊂ G is weakly closed in L if every conjugate of V , gV g^{−1} by an element of G which is contained in L is already conjugate to V by an element of L.

Theorem 1.1: For M_{23} the images of restriction in H^{∗}(E; _{2}) and H^{∗}(F; _{2}) are the entire invariant subrings. The image of restriction in H^{∗}(V; _{2}) is

2[d_{4}^{2}, d_{6}, d_{7}](1, d_{4}d_{6}, d_{4}d_{7})

where the d_{i} are the Dickson generators in dimension i. The “multiple image classes” are given as follows

(1) The polynomial subalgebra _{2}[d̄_{8}, d̄_{12}] injects diagonally into all three, the generators mapping as d_{8} ↦ (d_{8}, d_{8}, d_{4}^{2}), d_{12} ↦ (d_{12}, d_{12}, d_{6}^{2}),

(2) There are elements

LM ∈ H^{6}(F; _{2})^{GL_{2}( _{4}):2} , Sq^{4}(LM) = pq ∈ H^{10}(F; _{2})^{GL_{2}( _{4}):2}

so (0, LM, d_{6}), (0, pq, d_{4}d_{6}) are in the image of \overline{res}^{∗}. In particular, the non-multiple image classes in H^{∗}(V; _{2}) are exactly the elements contained in the ideal (d_{7}) ⊂ 2[d_{4}, d_{6}, d_{7}].

Finally, the kernel of \overline{res}^{∗} is _{2}[d̄_{8}, d̄_{12}](a_{7}, a_{11}).

(The situation for (1.1(2)) is identical to that for H^{∗}(M_{22}; _{2}).)

Remark 1.2: H^{∗}(E; _{2})^{A_{7}} has Poincaré series

(1.3)  \frac{1 + x^{18} + x^{20} + x^{21} + x^{24} + x^{25} + x^{27} + x^{45}}{(1 − x^{8})(1 − x^{12})(1 − x^{14})(1 − x^{15})}

and is Cohen-Macaulay over the Dickson algebra _{2}[d_{8}, d_{12}, d_{14}, d_{15}] with generators 1, b_{18}, b_{20} = Sq^{2}(b_{18}), b_{21} = Sq^{1}(b_{20}), b_{24}, b_{25} = Sq^{1}(b_{24}), b_{27} = Sq^{2}(b_{25}) and b_{45} from [AM2]. In particular, since the multiple image classes in H^{∗}(E; _{2}) modulo the classes which live only on this summand is _{2}[d_{8}, d_{12}](1, w̄_{3}) from [AM3], it follows easily that the only possible multiple image classes among the A_{7} invariants are _{2}[d_{8}, d_{12}].

Remark 1.4: In [AM1] we determined the ring of invariants H^{∗}(F; _{2})^{SL_{2}(4):2} and in §3 we extend that calculation to determine the GL_{2}(4): 2 invariants. The result is that the Poincaré series has the form p(x)/q(x) where q(x) is the polynomial

q(x) = (1 − x^{10})(1 − x^{12})(1 − x^{15})(1 − x^{24}),

and p(x) is

1 + x^{6} + 2x^{8} + x^{9} + x^{11} + 2x^{12} + x^{13} + 3x^{14} + 2x^{15} + 3x^{16}
+ 3z^{17} + 2x^{18} + 2x^{19} + 4x^{20} + 3x^{21} + 4x^{22} + 4x^{23} + 4x^{24} + 4x^{25}
+ 4x^{26} + 4x^{27} + 5x^{28} + 5x^{29} + 4x^{30} + 4x^{31} + 4x^{32} + 4x^{33} + 4x^{34}
+ 4x^{35} + 3x^{36} + 4x^{37} + 2x^{38} + 2x^{39} + 3x^{40} + 3x^{41} + 2x^{42} + 3x^{43}
+ x^{44} + 2x^{45} + x^{46} + x^{48} + 2x^{49} + x^{51} + x^{57}

Expanding out into a Taylor series we obtain for the first few terms the expansion

Corollary 1.5: The Poincaré series for the invariants _{2}[x_{1}, x_{2}, x_{3}, x_{4}]^{GL_{2}(4):2} has Taylor series of the form

1 + x^{6} + 2x^{8} + x^{9} + x^{10} + x^{11} + 3x^{12} + x^{13} + 3x^{14} + 3x^{15} + 4x^{16} + 3x^{17} + 5x^{18} + · · · .

But, comparing with the answer for M_{22} where

H^{∗}(F; _{2})^{S_{5}} = _{2}[w̄_{3}, γ_{5}, d_{8}, d_{12}](1, LM, pq, ⋯),

we see that LM is in the GL_{2}(4): 2 invariant subring and hence remains a multiple image class with image (0, LM, d_{6}) (as it is for M_{22}) as does Sq^{4}(LM) = pq. This accounts for all possible remaining multiple image classes. Additionally, we have

\begin{aligned}H^{i}(M_{23}; _{2}) & = 0, for i ≤ 5 \\ H^{i}(M_{23}; )_{(2)} & = 0, for i ≤ 6 \end{aligned}

where A_{(p)} denotes the p-primary part of the abelian group A.

The analysis at the remaining primes

For each of the odd primes dividing the order of M_{23} we have that Syl_{p}(M_{23}) is abelian. For the cohomology calculation in this case recall the result of R. Swan, [S],

Lemma 1.6: Let G be a finite group so that Syl_{p}(G) is abelian, then

H^{∗}(G; _{p}) = H^{∗}(Syl_{p}(G); _{p})^{W_{G}(Syl_{p}(G))}.

The analysis at 3. From [J], N(Syl_{3}(M_{23})) = 3^{2}: Syl_{2}(GL_{2}(3)) and C(3^{2}) = 3^{2}. We have that Syl_{2}(GL_{2}(3)) = SD_{16} the semi-dihedral group with presentation

{x, y | x^{8} = y^{2} = 1, yxy = x^{3}},

and

im(res^{∗}: H^{∗}(M_{23}; _{3})⟶H^{∗}(3^{2})) = { _{3}[X̄, Ȳ] ⊗ E(ē, f̄)}^{SD_{16}} .

To obtain this ring of invariants we proceed as follows. First tensor with _{9}, and choose a new basis X, Y, e, f, so x(e) = ζe, x(f) = ζ^{3}f, x(X) = ζX, and x(Y) = ζ^{3}Y where ζ is a primitive 8^{th} root of unity in _{9}. Also y interchanges X, Y, and e, f.

The invariants under x in _{9}[X, Y] ⊗ E(e, f) are given as follows

9[X^{8}, Y^{8}] (1, (XY)^{2}, X^{5}Y, XY^{5}, (XY)^{4},
e{XY^{2}, X^{4}Y, Y^{5}, X^{7}, X^{3}Y^{4}},
f{X^{2}Y, XY^{4}, X^{5}, Y^{7}, X^{4}Y^{3}},
ef{XY, X^{4}, Y^{4}, X^{3}Y^{3}, X^{6}Y^{2}, X^{2}Y^{6}})

and from this the structure of the ring H^{∗}(3^{2}; _{3})^{SD_{16}} can be easily read off.

Clearly, the first element fixed under x occurs in dimension 6, efXY, but the first element which is also fixed under y is eXY^{2} + fX^{2}Y in dimension 7. The analysis at 5. The normalizer of Syl_{5}(M_{23}) = /5 in M_{23} is the extension 5: 4 ⊂ A_{7}, and H^{∗}( /5; _{5})^{/4} = E(eb^{3}) ⊗ _{5}[b^{4}], with its lowest dimension generator occuring in dimension 7. The analysis at 7. The normalizer of Syl_{7}(M_{23}) = /7 in M_{23} is the extension 14: 3 and

H^{∗}( /7; _{7})^{/3} = E(eb^{2}) ⊗ _{7}[b^{3}]

with lowest dimensional generator occuring in dimension 5. The analysis at 11. The normalizer here is 11: 5 and

H^{∗}( /11; _{11})^{/5} = E(eb^{4}) ⊗ _{11}[b^{5}]

with lowest dimension generator occuring in dimension 9. The analysis at 23. The normalizer here is 23: 11 and

H^{∗}( /23; _{23})^{/11} = E(eb^{10}) ⊗ _{23}[b^{11}]

with lowest dimension generator occuring in dimension 21.

Taken together these results prove that H^{i}(M_{23}; ) = 0 for i ≤ 5 while

H^{6}(M_{23}; ) = /7 and H^{7}(M_{23}; ) = /2.

In particular, this shows, as asserted, that M_{23} provides a counterexample to the well known conjecture that if G is a finite group with H_{i}(G; ) = 0 for i = 1, 2, 3 then G = {1}.

Remark 1.7: The 2-Sylow subgroup of M_{23} is also Syl_{2}(M^{c}L), the McLaughlin sporadic group. Moreover, E and F remain non-conjugate in M^{c}L, though V becomes conjugate to a subgroup of E or F. Also, N_{M^{c}L}(E) ≅ N_{M^{c}L}(F) ≅ 2^{4}: A_{7}. Since E and F are weakly closed in Syl_{2}(M^{c}L) ⊂ M^{c}L it follows that the image of restriction res^{∗}: H^{∗}(M^{c}L; _{2})⟶H^{∗}(2^{4}; _{2}) is the entire invariant subring _{2}[x_{1}, x_{2}, x_{3}, x_{4}]^{A_{7}} for each of E and F. Additionally, the kernel of restriction

\overline{res}^{∗}: H^{∗}(M^{c}L; _{2})⟶H^{∗}(E; _{2})^{A_{7}} ⊕ H^{∗}(F; _{2})^{A_{7}}

is _{2}[d̄_{8}, d̄_{12}](a_{7}, a_{11}). But the determination of the exact multiple image classes is much more complex here than for M_{22} and M_{23}. This difficulty is resolved in recent joint work with A. Adem, where we first use our knowledge of H^{∗}(SU_{4}(3); _{2}) to obtain some low dimensional information about H^{∗}(P SU_{4}(3); _{2}). Since P SU_{4}(3) is a maximal, odd index subgroup of M^{c}L, this, together with our knowledge of H^{∗}(M_{22}; _{2}) determines the multiple image classes and hence, H^{∗}(M^{c}L; _{2}).

Remark 1.8: Another group which is closely related to the groups above is the sporadic group Ly with Syl_{2}(Ly) = Syl_{2}(M_{23}): 2 where the action of the new element of order two exchanges E, F. Since 3 · M^{c}L: 2 ⊂ Ly the knowledge we have of H^{∗}(M^{c}L; _{2}) leads to considerable information about H^{∗}(Ly; _{2}). However the complete determination of H^{∗}(Ly; _{2}) seems to depend on first, understanding more about the cohomology of the non-trivial central extension 2 · A_{11}, and second, relating these groups to the groups H^{∗}(3 · M^{c}L: 2; _{2}) which we now understand.

I would like to thank A. Adem for useful conversations. In particular he pointed out the reference [G].

§2. The structure of H^{∗}(M_{23}; _{2})

From [Co] we know that the normalizer of F in M_{23} has the form F: ((3 × A_{5}): 2) and we must determine the explicit action of (3 × A_{5}): 2 on F.

Lemma 2.1: Let g be the automorphism of GL_{2}( _{4}) which acts by x ↦ x^{2} on the coefficients in each matrix, then the action extends to an action on the affine group Aff_{2}( _{4}) = ( _{4})^{2}: GL_{2}( _{4}) and N_{M_{23}}(F) ≅ Aff_{2}( _{4}): 2 under this action.

Proof: SL_{2}( _{4}) = A_{5} and GL_{2}( _{4}) = SL_{2}( _{4}) × /3. We have that the image of 2(SL_{2}( _{4})) in M_{4}( _{2}) corresponding to the embedding SL_{2}( _{4}) ⊂ GL_{4}( _{2}) obtained by forgetting the _{4} structure is equal to the image of _{4}(SL_{2}( _{2})) which is M_{2}( _{4}). Hence the centralizer of SL_{2}( _{4}) is the centralizer of M_{2}( _{4}) which is _{4}, so there are only two actions of GL_{2}( _{4}) on ( _{4})^{2} extending the usual action of SL_{2}( _{4}), the usual one, and the one where the element of order three acts trivially. On the other hand, from [J, Lemma 9] there is only one conjugacy class of elements of order 3 in M_{23} and the centralizer of ⟨3⟩ has the form 3 × A_{5}. Hence the action cannot be the trivial one and the lemma follows.

In M_{22} the normalizers are E: A_{6} and F: S_{5}, with intersection F: S_{4} whenever E and F are contained in the same 2-Sylow subgroup of M_{22}. For M_{23} we have

Lemma 2.2: If representatives for E and F are chosen in the same Sylow subgroup of M_{23} then N(E) ∩ N(F) = F: ((3 × A_{4}): 2) where

(1) Syl_{2}(F: (3 × A_{4})) = Syl_{2}(L_{3}(4)) is the span ⟨E, F ⟩ ⊂ Syl_{2}(M_{23}), where L_{3}(4) = M_{21} ⊂ M_{22}.

(2) The extending element of order two acts on the 3 so 3: 2 = S_{3} and A_{4}: 2 = S_{4}.

(Indeed, Syl_{2}(SL_{2}( _{4})) fixes an _{4} ⊂ _{4}^{2} which we denote W, and consequently a representative for E is given as ⟨W, Syl_{2}(SL_{2}( _{4}))⟩. Since the new element of order three centralizes SL_{2}( _{4}) it must normalize E so it is contained in the intersection and the lemma follows.)

We are now able to characterize H^{∗}(M_{23}; _{2}) as a subring of H^{∗}(M_{22}; _{2}). This is the main result of this section.

Corollary 2.3: H^{∗}(M_{23}; _{2}) ⊂ H^{∗}(M_{22}; _{2}) is the set of α ∈ H^{∗}(M_{22}; _{2}) which restrict to H^{∗}(⟨E, F ⟩; _{2})^{3^{2}:2}.

Proof: There is a decomposition of M_{23} into double cosets of M_{22},

M_{23} = M_{22} ⊔ M_{22}T M_{22}

where T is the element of order three above, and T M_{22}T^{−1} ∩ M_{22} = M_{21} ≅ L_{3}(4). Then an element α ∈ H^{∗}(M_{22}; _{2}) is in the image of restriction from H^{∗}(M_{23}; _{2}) if and only if T^{∗}res^{∗}(α) = res^{∗}(α) in the (non-commutative) diagram

M_{21} ⟶^{T} T M_{21}T^{−1}
↓i  ↓i

M_{22} ⟶^{id}  M_{22}

from the Cartan-Eilenberg double coset formula. However, by our choice of T as the coset representative, T normalizes Syl_{2}(L_{3}(4)) = ⟨E, F ⟩, and it follows that α satisfies the restriction condition above if and only if res^{∗}(α) ∈ H^{∗}(Syl_{2}(L_{3}(4)); _{2}) is invariant under T.

§3. Detailed results

We see that the structure of H^{∗}(M_{23}; _{2}) is determined by giving the restriction image

\overline{res}^{∗}: H^{∗}(M_{23}; _{2})⟶H^{∗}(E; _{2})^{A_{7}} ⊕ H^{∗}(F; _{2})^{GL_{2}(4):2} ⊕ H^{∗}(V; _{2})^{GL_{3}(2)} and determining which elements in Ker(\overline{res}^{∗}) upon restriction to H^{∗}(⟨E, F ⟩; _{2}) are invariant under the action of the new element of order three.

We begin with the determination of the image of \overline{res}^{∗} for E and F.

From [AM3] both E and F are weakly closed in Syl_{2}(M_{22}) ⊂ M_{22}. Hence, they are also weakly closed in Syl_{2}(M_{22}) ⊂ M_{23} so we have

a. The restriction map H^{∗}(M_{23}; _{2})⟶H^{∗}(F) has image exactly the ring of invariants

2[x_{1}, …, x_{4}]^{GL_{2}(4):2}.

b. The restriction map H^{∗}(M_{23}; _{2})⟶H^{∗}(E) has image exactly the ring of invariants

2[x_{1}, …, x_{4}]^{A_{7}}.

In [AM2] the A_{7} invariants are studied and the answer has been described in (1.2). The ring of invariants _{2}[x_{1}, …, x_{4}]^{GL_{2}( _{4})} does not appear in the literature to our knowledge so we determine it now using the results of [AM1]. There we determine the ring _{4} ⊗_{_{2}} 2[x_{1}, …, x_{4}]^{SL_{2}(4)}. It is given as follows.

Theorem 3.1: _{4}[A, B, C, D]^{SL_{2}(4)} is generated by 9 elements, four polynomial generators, which generate a polynomial algebra P = _{4}[D_{12}(1), D_{12}(2), p, q], where

p = A^{4}C + AC^{4}, q = B^{4}D + BD^{4},
D_{12}(1) = A^{12} + A^{9}C^{3} + A^{6}C^{6} + A^{3}C^{9} + C^{12},
D_{12}(2) = B^{12} + B^{9}D^{3} + B^{6}D^{6} + B^{3}D^{9} + D^{12},

and five other generators

M = A^{2}D + C^{2}B, L = AD^{2} + CB^{2},
C = C^{4}D^{4} + C^{4}B^{3}D + AC^{3}B^{4} + A^{2}C^{2}B^{2}D^{2} + A^{3}CD^{4} + A^{4}BD^{3} + A^{4}B^{4},
D = AD^{8} + CB^{8}, E = A^{8}D + C^{8}D,

and the entire ring of invariants is free over P on 60 generators. A complete table of these generators is given as

3.2

Dimension  Module generators
0  1
3  L, M
6  L^{2}, LM, M^{2}
8  C
9  D, L^{3}, L^{2}M, LM^{2}, M^{3}, E
11  LC, MC
12  LD, MD, L^{4}, L^{3}M, L^{2}M^{2}, LM^{3}, M^{4}, LE, ME
14  LMC
15  L^{2}D, LMD, M^{2}D, L^{4}M, L^{3}M^{2}, L^{2}M^{3}, LM^{4}, L^{2}E, LME, M^{2}E
16  C^{2}
18  L^{3}D, L^{2}MD, LM^{2}D, M^{3}D, DE, L^{3}E, L^{2}ME, LM^{2}E, M^{3}E
19  LC^{2}, MC^{2}
21  L^{2}M^{2}D, M^{4}D, L^{4}M^{3}, L^{3}M^{4}, L^{4}E, L^{2}M^{2}E
22  LMC^{2}
24  L^{5}M^{3}, L^{4}M^{4}, L^{3}M^{5}
27  L^{5}M^{4}, L^{4}M^{5}
30  L^{5}M^{5}

At this stage /3 can be chosen to act via T A = ζA, T B = ζ^{2}B, while T fixes C, D. Thus the two 12 dimensional polynomial generators and C are fixed by T, but p, L, and D are multiplied by ζ while q, M and E are multiplied by ζ^{2}. Consequently a direct calculation gives the following refinement of (3.2)

3.3

Dim  Fixed  ζ  ζ^{2}
3  L  M
6  LM  M^{2}  L^{2}
8  C
9  L^{3}, M^{3}  D, L^{2}M  E, LM^{2}
11  LC  MC
12  MD, L^{2}M^{2}, LE  L^{4}, LM^{3}, ME  M^{4}, LD, L^{3}M
15 L^{2}D, L^{4}M, LM^{4}, M^{2}E LMD, L^{3}M^{2}, L^{2}E M^{2}D, L^{2}M^{3}, LME
14  LMC
16  C^{2}
18  LM^{2}D, DE, L^{2}ME L^{3}D, M^{3}D, LM^{3}E L^{3}E, M^{3}E, L^{2}MD
19  LC^{2}  MC^{2}
21  M^{4}D, L^{4}E  L^{2}M^{2}D, L^{4}M^{3}  L^{2}M^{2}E, L^{3}M^{4}
22  LMC^{2}
24  L^{4}M^{4}  L^{3}M^{5}  L^{5}M^{3}
27  L^{5}M^{4}  L^{4}M^{5}
30  L^{5}M^{5}

We first look at the new /3 invariants. We can write

P = _{4}[D_{12}(1), D_{12}(2), pq, p^{3} + q^{3}](1, p, q, p^{2}, q^{2}, q^{3}).

Let us write Q = _{4}[D_{12}(1), D_{12}(2), pq, q^{3} + q^{3}]: then the invariants have the form

3.4  Q(1, q^{3})W_{1} ⊕ Q(p, q^{2})W_{ζ^{2}} ⊕ Q(q, p^{2})W_{ζ}

where W_{1} is the set consisting of 1 and the elements in the first column of the table above, W_{ζ} is the set consisting of the elements in the second column above and W_{ζ^{2}} is the set consisting of the elements in the third column above. From this it is immediate that the Poincaré series for the GL_{3}(4) invariants has denominator

(1 − x^{12})^{2}(1 − x^{10})(1 − x^{15})

and numerator

1+x^{6} + 3x^{8} + 2x^{9} + 2x^{11} + 3x^{12} + 2x^{13} + 5x^{14} + 5x^{15} + 5x^{16}
+ 6x^{17} + 3x^{18} + 4x^{19} + 6x^{20} + 5x^{21} + 7x^{22} + 7x^{23} + 5x^{24}
+ 6x^{25} + 4x^{26} + 3x^{27} + 6x^{28} + 5x^{29} + 5x^{30} + 5x^{31} + 2x^{32}
+ 3x^{33} + 2x^{34} + 2x^{36} + 3x^{37} + x^{39} + x^{45}

Clearly y exchanges the second and third summands in (3.4) and we can decompose the first summand into

\begin{aligned}Q′ & (1, LM, C, (LM)^{2}, LMC, C^{2}, DE, LMC^{2}, (LM)^{4}, (LM)^{5})⊕ \\ & Q(1, q^{3})(L^{3}, M^{3}, MD, LE, L^{2}D, L^{4}M, LM^{4}, M^{2}E, LM^{2}D, L^{2}ME, M^{4}D, L^{4}E) \end{aligned}

where Q′ = _{4}[D_{12}(1) + D_{12}(2), D_{12}(1)D_{12}(2), pq, q^{3} + q^{3}](1, D_{12}(1)q^{3} + D_{12}(2)p^{3}). In particular, the first generator LM, occuring in dimension 6 is invariant under the action of the generator of the /2 which interchanges L, M, D, E, as well as the two polynomial generators of dimensions 5 and 12 respectively. It follows that the lowest dimensional generator for the ring of invariants under the full group GL_{2}(4): 2 is LM in dimension 6. But we can determine the entire ring of invariants in a completely direct manner. Consequently the Poincaré series for this ring of invariants has numerator

(1+x^{27})(1 + x^{6} + x^{8} + x^{12} + x^{14} + x^{16} + x^{18} + x^{24} + x^{30})
+ (1 + x^{12})(1 + x^{15})(x^{9} + x^{12} + 2x^{15} + x^{18} + x^{21})

+ (1 + x^{12})(x^{3} + x^{6} + 2x^{9} + x^{11} + 3x^{12} + 3x^{15} + 3x^{18} + x^{19} + 2x^{21} + x^{24} + x^{27})

and denominator

(1 − x^{10})(1 − x^{12})(1 − x^{15})(1 − x^{24}).

The numerator above is expanded out in (1.4).

With the action of the new element of order 3 on the elements in Syl_{2}(L_{3}(3)) as determined above we can now consider the invariants in the kernel of \overline{res}^{∗}.

Lemma 3.5: H^{∗}(⟨E, F ⟩; _{2})^{3^{2}:2} ∩ res^{∗}(Ker(\overline{res}^{∗})) = _{2}[d̄_{8}, d̄_{12}](a_{7}, a_{11}).

Proof: This comes from the discussion of the Kernel of restriction to H^{∗}(E; _{2}) ⊕ H^{∗}(F; _{2}) in H^{∗}(⟨E, F ⟩; _{2}) at the beginning of §3 of [AM1], in particular the expansion on the top of page 197,

4[v^{3}, w^{3}](1, vw, v^{2}w^{2})(AF, BE, AFM, LM + ABC^{2}D^{2})
⊕ _{4}[v^{3}, w^{3}](v, v^{2}w, w^{2})FA^{2}
⊕ _{4}[v^{3}, w^{3}](w, vw^{2}, v^{2})AF^{2}.

The elements AFM and LM + ABC^{2}D^{2} are in the image of restriction to elementary two groups in Syl_{2}(M_{22}), so they can be ignored. Also, v = C^{4}, w = D^{4}, and our choice of T forces it to act as non-trivially on F so that FA^{2} is invariant. Finally, according to [AM3], the element v^{2}wFA^{2} + vw^{2}AF^{2} is not in the image of restriction from H^{∗}(M_{22}; _{2}).

It remains to consider the group V. As a subgroup of M_{22} we showed in [AM3] that it is weakly closed in Syl_{2}(M_{22}). As a corollary of the result above we have that this remains true in M_{23}.

Lemma 3.6: The extremal 2^{3}, V in M_{22} is not conjugate to any subgroup of E or F in M_{23}. In particular V ⊂ Syl_{2}(M_{22}) remains weakly closed in M_{23} and the image of restriction from H^{∗}(M_{23}) in H^{∗}(V) is _{2}[d_{6}, d_{7}, d_{4}^{2}](1, d_{4}d_{6}, d_{4}d_{7}).

Proof: The element LM satisfies the condition that it restricts to (0, LM, d_{6}) in H^{6}(E)⊕ H^{6}(F)⊕H^{6}(V) from [AM3], and from the weak closure property of E, F, it must be in the image of restriction from H^{∗}(M_{23}). As a consequence Sq^{1}(LM) which restricts to (0, 0, d_{7}) must also be in the image from H^{∗}(M_{23}). Now, if V were conjugate to a subgroup of E or F, since there are no A_{7} or GL_{2}(4): 2 invariants in dimension 7, it would follow that Sq^{1}(LM) would restrict to zero in all three groups which is impossible. This proves the first statement. The weak closure now follows from weak closure in M_{22}, and the cohomology calculation follows from the calculation in [AM3] and the fact that N(V) = V: GL_{3}(2), already in M_{22}.

Now, between H^{∗}(E; _{2}) and H^{∗}(F; _{2}) there can remain only _{2}[d_{8}, d_{12}] as multiple image classes, and the multiple image classes between H^{∗}(V; _{2}) and H^{∗}(F; _{2}) were determined in [AM3]. Thus, we have an effective determination of H^{∗}(M_{23}; _{2}).

§4. The map H^{∗}(S_{23})⟶H^{∗}(M_{23})

In [Go], pg. 79, explicit generators are given for M_{22} ⊂ S_{22} and M_{23} ⊂ S_{23} so that we have the commutative diagram

M_{22} ↪ S_{22}
↓in  ↓in
M_{23} ↪ S_{23}.

In terms of these generators a computer calculation was done to identify the centralizer of the element

(1, 16)(2, 13)(3, 5)(7, 12)(9, 10)(14, 15)(17, 20)(18, 19)

in M_{22} and thereby get an explicit determination of Syl_{2}(M_{22}) ⊂ S_{22}. The result, as far as we need it now, was to determine explicit generators for that the two subgroups 2^{4} ⊂ Syl_{2}(M_{22}). The first is generated by the following four elements

4.1  \begin{matrix}(1, 18)(2, 12)(3, 17)(5, 20)(7, 13)(9, 15)(10, 14)(16, 19) \\ (1, 12)(2, 18)(3, 15)(5, 14)(7, 16)(9, 17)(10, 20)(13, 19) \\ (1, 16)(2, 13)(3, 5)(7, 12)(9, 10)(14, 15)(17, 20)(18, 19) \\ (1, 17)(2, 15)(3, 18)(5, 19)(7, 10)(9, 12)(13, 14)(16, 20), \end{matrix}

and the second is generated by the elements

4.2  \begin{matrix}(1, 16)(2, 13)(6, 21)(7, 18)(8, 11)(12, 19)(14, 20)(15, 17) \\ (1, 13)(2, 16)(3, 9)(5, 10)(6, 21)(7, 12)(8, 11)(18, 19) \\ (1, 16)(2, 13)(3, 5)(7, 12)(9, 10)(14, 15)(17, 20)(18, 19) \\ (1, 13)(2, 16)(3, 5)(6, 11)(8, 21)(9, 10)(14, 20)(15, 17). \end{matrix}

We now determine the conjugacy classes of the resulting groups in S_{20} ⊂ S_{23}. The reason this is useful is that it is fairly easy to determine the image of the restriction map from H^{∗}(G; _{2}) to H^{∗}(H; _{2}), H ⊂ G, if G is 2-elementary, and since the composition

H^{∗}(S_{23}; _{2})⟶^{res^{∗}}L⟶^{res^{∗}}H^{∗}(D)

factors through the restriction to H^{∗}(M_{23}; _{2}) if D is one of the two 2-elementaries E, V, or F and L is a maximal 2-elementary in S_{23} which contains it this gives a way of understanding something about the cohomology map from H^{∗}(S_{23}; _{2}) to H^{∗}(M_{23}; _{2}).

To begin we recall the classification of the conjugacy classes of maximal elementary 2-subgroups of S_{n}. First, there is always the regular representation 2^{n} ↪ S_{2^{n}} which gives us a maximal elementary subgroup which we write V_{n}. The regular representation of 2^{2} in S_{4} is the Klein group K, and the regular representation of 2^{n} is seen to be given explicitly via the following composition

2^{n−1} × 2⟶^{Δ(reg)×id}S_{2^{n}} ≀ 2 ↪ S_{2^{n}},

where Δ is the diagonal map 2^{n−1}⟶2^{n−1} × 2^{n−1}, where Δ(x) = (x, x). Then, if E ⊂ S_{2n} is a maximal 2-elementary, it can be written uniquely, up to conjugacy, as the product V_{i_{1}} × V_{i_{2}} × ⋯ × V_{i_{r}} where 2^{i_{1}} + 2^{i_{2}} + ⋯ + 2^{i_{r}} = 2n, ([AM4], Chapt VI). For example the two maximal subgroups of S_{4} are V_{1} × V_{1} and K.

Remark: The Weyl group of (V_{n})^{k} ⊂ S_{2^{n}k} is L_{n}(2) ≀ S_{k}. The image of restriction from H^{∗}(S_{2^{n}}; _{2}) to H^{∗}(V_{n}; _{2}) is always the Dickson algebra; a polynomial algebra of the form

D_{n} = _{2}[d_{2^{n−1}}, d_{2^{n−1}+2^{n−2}}, …, d_{2^{n}−1}],

and the image of H^{∗}(S_{2^{n}k}; _{2}) in H^{∗}(V_{n}^{k}; _{2}) is the ring of invariants

(\underbrace{D_{n} ⊗ D_{n} ⊗ ⋯ ⊗ D_{n}}_{k−times})^{S_{k}} .

As an example D_{2} = _{2}[a^{2} + b^{2} + ab, a^{b} + ab^{2}] where a, b ∈ H^{1}(V_{2}; _{2}) are any pair of polynomial generators. Incidently, D_{n} is exactly H^{∗}(V_{n}; _{2})^{L_{n}(2)}, so these cohomology images are as simple as it would be possible for them to be.

We now return to the groups determined by the elements in (4.1) and (4.2). Note that the first two elements in (4.1) determine a subgroup of K × K × K × K ⊂ S_{16} where K is the Klein group contained in S_{4}, and the four S_{4}’s respectively permute the 4-tuples (1, 2, 12, 18), (3, 5, 17, 20), (7, 13, 16, 19), and (9, 10, 14, 15). Then the third element acts to exchange the sets (1, 2, 12, 18) with (7, 13, 16, 19), and simultaneously the other two sets, and finally the forth element acts to exchange {(1, 2, 12, 18), (7, 13, 16, 19)} with its compliment. Consequently, this give the regular representation of 2^{4} ⊂ S_{16}.

For the second group, by looking at the first two elements we get that the subgroup they generate is contained in the product of five copies of K, K × K × K × K × K ⊂ S_{20}. In fact, we get the following table where the elements are now rearranged so that the part involving each K is separated out:

4.3 | \begin{matrix}(1, 16)(2, 13) \\ (1, 13)(2, 16) \\ (1, 16)(2, 13) \\ (1, 13)(2, 16) \end{matrix} | \begin{matrix}(6, 21)(8, 11) \\ (6, 21)(8, 11) \\ (6, 11)(8, 21) \end{matrix} | \begin{matrix}(7, 18)(12, 19) \\ (7, 12)(18, 19) \\ (7, 12)(18, 19) \end{matrix} | \begin{matrix}(3, 9)(5, 10) \\ (3, 5)(9, 10) \\ (3, 5)(9, 10) \end{matrix} | \begin{matrix}(14, 20)(15, 17) \\ (14, 15)(17, 20) \\ (14, 20)(15, 17) \end{matrix} | .

The Weyl group of V_{4} in S_{16} is L_{4}(3), while the Weyl group of K^{5} in S_{20} is L_{2}(2) ≀ S_{5}. From this it follows that the subgroup of M_{23} generated by the elements in (4.1) is E and the subgroup generated by the elements in (4.2) is F.

Now we determine the restriction map from

H^{∗}(S_{23}) to H^{∗}(E)^{A_{7}} ⊕ H^{∗}(F)^{GL_{2}(4):2}

from the commutative diagrams:

4.4  \begin{matrix}E & ↪ & M_{23} \\ ↓≅ & & ↓in \\ V_{4} & ↪ & S_{23} \end{matrix}

and

4.5  \begin{matrix}F & ↪ & M_{23} \\ ↓in & & ↓in \\ K^{5} & ↪ & S_{23} \end{matrix}

since the images of the restriction maps res^{∗} are known for V_{n} and K^{n} from [AM4], pp. 182-204.

In the case of E, the left vertical arrow in (4.4) is an isomorphism and the image of res^{∗} from H^{∗}(S_{23}) in H^{∗}(V_{4}) is exactly the Dickson algebra, _{2}[d_{8}, d_{12}, d_{14}, d_{15}]. However, for the group F the situation is more subtle. The cohomology map

H^{∗}(K^{5})⟶H^{∗}(F)

is determined on generators as follows:

\begin{matrix}a_{1} ↦ e_{1} + e_{3}, & b_{1} ↦ e_{2} + e_{4} \\ a_{2} ↦ e_{1} + e_{2} & b_{2} ↦ e_{4} \\ a_{3} ↦ e_{1} & b_{3} ↦ e_{2} + e_{3} \\ a_{4} ↦ e_{2} & b_{4} ↦ e_{3} + e_{4} \\ a_{5} ↦ e_{1} + e_{4} & b_{5} ↦ e_{3}, \end{matrix}

using the decomposition (4.3) above and choosing the first element appearing in the i^{th} column as dual to a_{i}, while the second distinct element is dual to b_{i}.

Theorem 4.6: The non-zero element in H^{6}(M_{23}; _{2}) is contained in the image of restriction from H^{∗}(S_{23}; _{2}). Specifically, it is the image of the symmetric sum Σ(d_{3} ⊗ d_{3}).

Proof: This was a direct exercise using MACSYMA. The expressions above in e_{1}, …, e_{4} for the a_{i} and b_{i} were substituted into the expression for the symmetric sum Σ_{i<j}(d_{3,i}d_{3,j}) in _{2}[d_{2,1}, …, d_{2,5}, d_{3,1}, …, d_{3,5}] and the result was

e_{1}e_{3}^{2}e_{4}^{3} + e_{1}^{2}e_{3}e_{4}^{3} + e_{1}e_{2}^{2}e_{4}^{3} + e_{1}^{2}e_{2}e_{4}^{3} + e_{2}e_{3}^{3}e_{4}^{2} + e_{1}e_{3}^{3}e_{4}^{2}
+ e_{2}^{3}e_{3}e_{4}^{2} + e_{1}^{3}e_{2}e_{4}^{2} + e_{2}^{2}e_{3}^{3}e_{4} + e_{1}^{2}e_{3}^{3}e_{4} + e_{2}^{3}e_{3}^{2}e_{4}
+ e_{1}^{3}e_{2}^{2}e_{4} + e_{1}e_{2}^{3}e_{3}^{2} + e_{1}^{3}e_{2}e_{3}^{2} + e_{1}^{2}e_{2}^{3}e_{3} + e_{1}^{3}e_{2}^{2}e_{3}+
e_{3}^{2}e_{4}^{4} + e_{2}^{2}e_{4}^{4} + e_{3}^{4}e_{4}^{2} + e_{2}^{4}e_{4}^{2} + e_{1}^{4}e_{4}^{2} + e_{2}^{2}e_{3}^{4} + e_{1}^{2}e_{3}^{4}
+ e_{1}^{2}e_{4}^{4} + e_{2}^{4}e_{3}^{2} + e_{1}^{4}e_{3}^{2} + e_{1}^{2}e_{2}^{4} + e_{1}^{4}e_{2}^{2}+
e_{1}e_{3}e_{4}^{4} + e_{1}e_{2}e_{4}^{4} + e_{2}e_{3}^{4}e_{4} + e_{1}e_{3}^{4}e_{4}
+ e_{2}^{4}e_{3}e_{4} + e_{1}^{4}e_{2}e_{4} + e_{1}e_{2}^{4}e_{3} + e_{1}^{4}e_{2}e_{3}+
e_{1}^{2}e_{2}e_{3}e_{4}^{2} + e_{1}e_{2}e_{3}^{2}e_{4}^{2} + e_{1}e_{2}^{2}e_{3}^{2}e_{4} + e_{1}^{2}e_{2}^{2}e_{3}e_{4},

which is non-zero and hence a representative for LM.

This particular choice for element to check was determined by homotopy theoretic considerations and will be explained further now. The inclusion M_{23} ⊂ S_{23} induces an inclusion of classifying spaces B_{M_{23}} ↪ B_{S_{23}}, and passing to Quillen plus constructions, a map

B_{M_{23}}^{+}⟶B_{S_{23}}^{+} ↪ B_{S_{∞}}^{+} = Q_{0}S^{0}

where Q_{0}S^{0} is the 0-component of the limit lim_{n↦∞}(Ω^{n}S^{n}). This space is well known to decompose as a product. First Q_{0}S^{0} decomposes as a product of its prime localizations

Q_{0}S^{0} ≃ (Q_{0}S^{0})_{2} × (Q_{0}S^{0})_{3} × (Q_{0}S^{0})_{5} × ⋯,

where π_{i}((Q_{0}S^{0})_{p}) = π_{i}^{s}(S^{0})_{p}, the p-primary part of the i-th stable homotopy group of spheres. Moreover, at each prime p there is a splitting

(Q_{0}S^{0})_{p} ≃ im(J)_{p} × Coker(J)_{p},

where im(J)_{p} is well understood and gives rise to the p-local part of the image of the J-homomorphism in homotopy. On the other hand, the space Coker(J)_{p} is 5-connected

and π_{6}(Coker(J)_{p}) = \{ \begin{matrix}H_{6}(Coker(J)_{2}; ) = /2 & for p = 2, \\ 0 & if p > 2. \end{matrix} Moreover, the element in

H_{6}(Q_{0}S^{0}; /2) which is the image of this class is dual to the element S(d_{3} ⊗ d_{3}). Thus, this class was really the only possiblility.

The spaces im(J)_{2} are identified by Quillen with the spaces lim_{n↦∞}(B_{O_{n}(3)}^{+}), and for each odd prime p, there is a prime q so that lim_{n↦∞}(B_{GL_{n}( _{q})}^{+} )_{p} = im(J)_{p}. Thus, the groups of (classical) Lie type are playing a basic role in the im(J) part of the homotopy of spheres. So it seems reasonable to suppose that the sporadic groups and the exceptional Chevellay groups should play an important role in the structure of the Coker(J) part of homotopy theory. This seems to be confirmed here for M_{23} and probably is a point of view which merits further study.

§5. The inclusion M_{12} ↪ S_{12}

The role that M_{12} should play in homotopy theory, from this point of view, seems to be involved with the theta invariant class σ^{2} ∈ π_{14}^{s}(S^{0}), see, e.g., [BW], [M]. It is known that π_{14}^{s}(S^{0}) = ( /2)^{2} with generators κ and σ^{2}, [T]. It is also known that κ is obtained from the G_{2}-invariant framing of G_{2}, and there is a 2-local fibration, X_{14}⟶B_{M_{12}}^{+}⟶B_{G_{2}} where the fiber is a 14-dimensional Poincaré complex with fundamental group mapping surjectively onto M_{12}, [BW], [M]. Thus, we would hope that there was a homomorphism M_{12}⟶S_{n} for some n which, on passing to homology, makes visible the connection between X_{14} and σ^{2}.

At this time we do not quite have the desired map, but it is relatively direct to extend the techniques above to discuss the homology maps induced from homomorphisms M_{12}⟶S_{n}. We illustrate this by describing the results for the usual inclusion, M_{12} ↪ S_{12} and the composite inclusion M_{12} ↪ M_{24} ↪ S_{24}.

The inclusion M_{12} ↪ S_{12}

There are three conjugacy classes of extremal 2-elementaries in M_{12}, and with respect to the usual embedding into S_{12}, given for example in [AMM2], we find that generators for the first, K, are

5.1  \begin{matrix}zz \\ [z, zz] \\ a^{2} \end{matrix} | \begin{matrix}(1, 11)(10, 12) \\ (1, 12)(10, 11) \end{matrix} | (2, 3) | \begin{matrix}(4, 7) \\ (4, 7) \end{matrix} | \begin{matrix}(5, 9) \\ (5, 9) \end{matrix} | \begin{matrix}(6, 8) \\ (6, 8) \\ (6, 8) \end{matrix} |

which shows that K ↪ V_{2} × (V_{1})^{4} ⊂ S_{12}. For the second, L, we have

5.2  \begin{matrix}zz a \\ [z, zz] \\ a^{2} \end{matrix} | \begin{matrix}1, 11)(10, 12) \\ (1, 12)(10, 11) \end{matrix} | \begin{matrix}(5, 8)(6, 9) \\ (5, 9)(6, 8) \\ (5, 9)(6, 8) \end{matrix} | \begin{matrix}(2, 7)(3, 4) \\ (2, 3)(4, 7) \end{matrix} |

which shows that L ↪ (V_{2})^{3} ⊂ S_{12}. Finally, the last group M gives

5.3  \begin{matrix}b[z, zz]a \\ zz a \\ a^{2} \end{matrix} | \begin{matrix}(1, 12)(10, 11) \\ (1, 11)(10, 12) \end{matrix} | \begin{matrix}(5, 7)(4, 9)(3, 6)(2, 8) \\ (5, 8)(6, 9)(2, 7)(3, 4) \\ (5, 9)(8, 6)(2, 3)(4, 7) \end{matrix} |

so M ↪ V_{2} × V_{3} ⊂ S_{12}. Additionally, in [AMM2] it was shown that restriction to these three subgroups detects H^{∗}(M_{12}; _{2}). Since H^{∗}(S_{n}; _{2}) is also detected by restriction to its 2-elementaries for all n, we need only study restriction to these three groups to entirely determine the image of restriction to H^{∗}(M_{12}; _{2}).

In [AMM1] the cohomology ring H^{∗}(S_{12}; _{2}) was determined, and it was shown that over the Steenrod algebra A(2) it has five generators σ_{1}, σ_{2}, σ_{3}, σ_{4}, and σ_{6}. Under restriction we get the following table

V_{2} × (V_{1})^{4}  (V_{2})^{3}  V_{2} × V_{3}
σ_{1}  1 ⊗ σ_{1}  0  0
σ_{2} d_{2} ⊗ 1 + 1 ⊗ σ_{2} S(d_{2} ⊗ 1 ⊗ 1) d_{2} ⊗ 1
σ_{3} d_{2} ⊗ σ_{1} + 1 ⊗ σ_{3} 0 0
σ_{4} d_{2} ⊗ σ_{2} + 1 ⊗ σ_{4} S(d_{2} ⊗ d_{2} ⊗ 1) 1 ⊗ d_{4}
σ_{6}  d_{2} ⊗ σ_{4}  d_{2} ⊗ d_{2} ⊗ d_{2} d_{2} ⊗ d_{4}

which, with the data above determines the images of restriction from H^{∗}(S_{12}; _{2}) in these three subgroups. First, σ_{1} restricts to zero in all three, and for the rest we have

5.4  \begin{aligned}& K & L & M \\ σ_{2} & 0 & h^{2} & m^{2} + h^{2} + mh \\ σ_{3} & μτ(μ + τ) & 0 & 0 \\ σ_{4} & d_{4} & d_{4} & d_{4} \\ σ_{6} & (μ^{2} + τ^{2} + μτ)d_{4} & d_{21}d_{22}d_{23} & (m^{2} + h^{2} + mh)d_{4} \end{aligned}

where d_{21} = h^{2} + μ^{2} + hμ, d_{22} = h^{2} + λ^{2} + λh, d_{23} = (h^{2} + μ^{2} + λ^{2} + hμ + hλ), and λ is dual to a^{2}, τ is dual to zz, μ is dual to [z, zz], and h, m, are dual to the two remaining generators.

In [AMM2] the groups H^{∗}(M_{12}; _{2}) were shown to have the form

5.5  _{2}[a_{2}, x_{3}, y_{3}, z_{3}, d_{4}, x_{5}, d_{6}, d_{7}]/R

where R is an explicitly given set of relations. Hence a_{2} = res^{∗}(σ_{2}), and only the generator y_{3} which restricts to (0, h^{3}, m^{3} + h^{2}m + h^{3}) in dimension 3 is missing from the image. In particular x_{5} is Sq^{2}(res^{∗}(σ^{3})), and d_{4} = (d_{4}, d_{4}, d_{4}) is also in the image of restriction. Finally, d_{6} = Sq^{2}(d_{4}), d_{7} = Sq^{1}(d_{6}).

The inclusion M_{12} ↪ M_{24} ↪ S_{24}

Out(M_{12}) = /2 and we can write Aut(M_{12}) = M_{12}: 2 realizing the outer automorphism ϕ as an element of order 2 in Aut(M_{12}). Moreover, the inclusion M_{12} ↪ M_{12}: 2 ↪ M_{24} embeds M_{12} by x ↦ (x, ϕ(x)) ⊂ S_{24}. On the other hand, by checking maximal subgroups in M_{12}: 2 in [Co], one sees directly that the action of ϕ on the two maximal 2-elementaries, K and L can be assumed to be trivial, while the action on M is nontrivial†. Consequently, ϕ^{∗} fixes all the generators of H^{∗}(M_{12}; _{2}) given in 5.5, except the generator y_{3} for which ϕ^{∗}(y_{3}) = y_{3} + Sq^{1}(a_{2}). In particular, this shows how to evaluate the cohomology map j^{∗}: H^{∗}(S_{24}; _{2})⟶H^{∗}(M_{12}; _{2}) which factors through H^{∗}(M_{24}; _{2}) according to the commutative diagram of groups

M_{12}  ⟶^{Δ} M_{12} × M_{12} ⟶^{1×ϕ} M_{12} × M_{12}
↓in  ↓in×in

M_{24}  ⟶^{in}  S_{24}  ⟵^{in} S_{12} × S_{12}.

In H^{∗}(S_{24}; _{2}) a cohomology generator which is dual to the Hurewicz image of σ^{2} is the class Q_{2,6} which restricts, in V_{2}^{6} to the generator given as the symmetric sum

S(d_{3} ⊗ d_{3} ⊗ d_{2} ⊗ d_{2} ⊗ d_{2} ⊗ d_{2}),

while it restricts to S(d_{3} ⊗ d_{3} ⊗ σ_{8}) in V_{2}^{2} × V_{1}^{8}. It restricts to 0 in V_{2}^{2} × V_{3}^{2}.

A quick calculation then shows that j^{∗}(Q_{2,6}) restricts to the class

((τ^{4}S + τ^{2}T + τS^{2})^{2}, (h^{4}S + h^{2}T + hS^{2})^{2}, 0)

in (H^{∗}(K), H^{∗}(L), H^{∗}(M)) where S = μ^{2}λ + μλ^{2}, T = μ^{4}λ + μλ^{4}. This element should be dual to the class of the fiber in the map B_{M_{12}}^{+}⟶B_{G_{2}}^{+}, though that hasn’t been verified yet.

Bibliography

† I would like to thank Steve Smith for several helpful notes which helped me understand the action of ϕ.

[AMM1] Adem, A., Maginnis, J., Milgram, R.J., Symmetric invariants and cohomology of groups, Math. Ann., 287(1990), 391-411.

[AMM2] Adem, A., Maginnis, J., Milgram, R.J., The geometry and cohomology of the Mathieu

group M_{12}, J. of Algebra, 139 (1992), 90-133.

[AM1] Adem, A., Milgram, R.J., A_{5}-invariants, the cohomology of L_{3}(4) and related extensions, Proc. Lond. J. Math., 66(1993), 187-224.

[AM2] Adem, A., Milgram, R.J., Invariants and cohomology of groups, Boll. S. Math. Mex., (to appear).

[AM3] Adem, A., Milgram, R.J., The geometry and cohomology of the Mathieu group M_{22}, (to appear).

[AM4] Adem, A., Milgram, R.J., The Cohomology of Finite Groups, Grundlehren der Math., 309, Springer-Verlag (1994).

[BW] Benson, D.J., Wilkerson, C.W., Finite simple groups and Dickson invariants, Proceedings of Conference in Honor of S. Gitler’s 60th Birthday, Contempory Math, to appear.

[Co] J. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Oxford University Press (1985).

[DW] Dwyer W.G., Wilkerson, C.W., A new finite loop space at the prime two, JAMS 6

(1993), 37-64.

[G] Giffen, C., Contributed Problems, Proceedings Northwestern University Homotopy Theory Conference, Contemporary Mathematics, 19(1982), 447.

[Go] Gorenstein, D., Finite Simple Groups, Plenum Press, (1982).

[J] Janko, Z., A characterization of the Mathieu simple groups. II, J. Algebra 9(1968), 20-41.

[M] Milgram, R.J., On the geometry and cohomology of the simple groups G_{2}(q) and 3D_{4}(q), II, Preprint, Stanford U., 1990.

[S] Swan, R., The p-period of a finite group, Ill. J. Math., 4(1960), 341-346.

[T] Toda, H., Composition Methods in Homotopy Groups of Spheres, Ann. of Math Studies 49. Princeton U. Press, 1962.

Department of Applied Homotopy Stanford University Stanford California, 94305

Current Address Department of Mathematics and Statistics University of New Mexico Albuquerque, N.M., 87131

e-mail: milgram@gauss.stanford.edu