The Cohomology of the Sporadic Groups J_{2} and J_{3}

Jon F. Carlson^{†}

University of Georgia, Athens, GA

John S. Maginnis

Kansas State University, Manhattan, KS

R. James Milgram^{†}

Stanford University, Stanford, CA

In this paper we determine the mod(2) cohomology rings of the two sporadic groups J_{2} and J_{3}. Our approach is to first use computer calculations in low dimensions to show that H^{∗}(Syl_{2}(J); 𝔽_{2}), for J either J_{2} or J_{3}, is detected by restriction to a single 2^{4} and three copies of (2 × Q_{8}). Then, using the known Weyl groups of 2^{4} in J_{2}, J_{3} and the fusions of the (2 × Q_{8})’s we easily determine the groups H^{∗}(J_{2}; 𝔽_{2}) and H^{∗}(J_{3}; 𝔽_{2}).

The reasons for considering these kinds of questions are threefold. First, there is the intrinsic interest in the details of the structure of any of the sporadic groups. Second, there are well known relations between the structure of the quotient of the cohomology ring H^{∗}(G; 𝔽_{p}) by the nil-radical, and the structure of the modular representations of G over the algebraic closure of 𝔽_{p}. These are given by varieties of modules and other data associated with the module categories and subcategories. Third, there is the connection with homotopy theory due to the fact that the space QS^{0} ≃ Ω^{∞}S^{∞} can be identified with B_{S_{∞}}^{+}, and the relation between the maximal subgroups of the S_{n} and the primitive actions of the simple groups via the O’Nan-Scott theorem, [LPS], [M2]. This last connection has been explored with some success for the Mathieu group M_{12}, [M2], and our work here represents the first step in a similar exploration of the implications of J_{2} and J_{3} in these areas. Moreover, at least for the third question, seeing the way in which the cohomology restricts to the various subgroups gives basic information on how the stable splitting of the classifying space will work, one of the main ingredients, so far in the applications to homotopy theory.

The Hall-Janko group J_{2} of order 604,800 and the Janko group J_{3} of order 50,232,960 are first characterized as simple groups having involution centralizer D_{8} ∗ Q_{8}: A_{5} and this subgroup is maximal in each. The Sylow 2-subgroup of J_{2}, which is equal to the Sylow 2-subgroup of J_{3}, Syl_{2}(J_{2}) = S_{J}, contains a unique copy of the central product D_{8} ∗Q_{8}, denoted 2_{−}^{1+4}, and also a unique copy of Syl_{2}(L_{3}(4)) which we denote 2^{2+4}. Their intersection is the commutator subgroup of S_{J}, S′_{J} ≅ Q_{8} × 2. (See e.g. the preliminary discussion in [Mag].)

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

The normalizer of 2^{2+4} in both J_{2} and J_{3} is a split extension 2^{2+4}: (S_{3} × 3), and the intersection of this normalizer with N_{J_{2}}(2_{−}^{1+4}) = 2_{−}^{1+4}: A_{5} is

N_{J_{2}}(2_{−}^{1+4}) ∩ N_{J_{2}}(2^{2+4}) = N_{J_{2}}(S_{J}) ≅ S_{J}: 3 ≅ 2_{−}^{1+4}: A_{4}.

The intersection is the same for J_{3}. Thus, since the involution centralizer above is maximal in both groups there are surjective homomorphisms of the amalgamated product W = 2_{−}^{1+4}: A_{5} ∗_{S_{J}:3} 2^{2+4}: (3 × S_{3}):

\begin{matrix}Φ: W & ⟶ J_{2} \\ Ψ: W & ⟶ J_{3}. \end{matrix}

Also both J_{2} and J_{3} contain a single conjugacy class of 2^{4}’s with

\begin{aligned}N_{J_{2}}(2^{4}) & = 2^{4}: (3 × A_{4}), \\ N_{J_{3}}(2^{4}) & = 2^{4}: (3 × A_{5}). \end{aligned}

In [Mag] the cohomology ring of S_{J} is determined:

Theorem [Mag]: The mod(2) cohomology of S_{J} is generated by three 1-dimensional classes w, x, y, two 2-dimensional classes α, β, two three dimensional classes L, M, one 4-dimensional class T_{4}, two 5-dimensional classes N_{1}, N_{2}, one 8-dimensional class d_{8} and one 10-dimensional class W. It is detected by restriction to the cohomology of 2_{−}^{1+4} and 2^{2+4}.

On the other hand, the cohomology of 2^{2+4} is determined in [AM1], following a calculation of D. Benson, (unpublished), while the cohomology of 2_{−}^{1+4} was determined by Quillen, [Q]. The restrictions of H^{∗}(S_{J}) to these cohomology rings are given explicitly in [Mag], (Lemma 3, and Theorem 8).

Our first main result in this note is the following.

Theorem A: Let G be a finite group given as the surjective image of W above, and assume the surjection restricts to give an isomorphism S_{J}⟶Syl_{2}(G). Then G contains a single conjugacy class of 2^{4}’s and a single conjugacy class of Q_{8} × 2’s. Moreover, its mod(2) cohomology is detected by restriction to these two groups.

Corollary : Let G be given as above, then H^{∗}(G; 𝔽_{2}) injects into H^{∗}(2^{2+4}).

In view of the results above and a close analysis of the ring of invariants in the radical of H^{∗}(2^{2+4}) we have the following. (See Section 6 for details.)

Corollary: Let G be given as above.

(a) The radical of H^{∗}(G; 𝔽_{2}) is a submodule of 𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11}), where all the cup products among k, a_{7} and a_{11} are zero.

(b) The subgroup 2^{4} ⊂ Syl_{2}(G) is weakly closed in G so the restriction of H^{∗}(G) to

H^{∗}(2^{4}) is

H^{∗}(2^{4})^{W_{G}(2^{4})} ∩ im (res^{∗}: H^{∗}(S_{J})⟶H^{∗}(2^{4})) .

(Here as in the rest of the paper, a subscript on a cohomology element denotes the degree of the element.)

The normalizer of 2^{4} ⊂ S_{J} is the 2^{2+4}. Let V ≅ 2^{2} ⊂ 2^{4} be the center of 2^{2+4}. Then there is an element of order 2, A ∈ S_{J}, which acts non-trivially on V while it permutes the two distinct 2^{4}’s contained in 2^{2+4}. Consequently,

im (res^{∗}: H^{∗}(S_{J})⟶H^{∗}(2^{4}))

is contained in the intersection H^{∗}(2^{4})^{2^{2}} ∩ L where L ⊂ H^{∗}(2^{4}) is the subring of elements which restrict to H^{∗}(V)^{⟨A⟩}.

The results of [Mag] show that this containment is proper. However, it turns out that the image of restriction from H^{∗}(J_{2}; 𝔽_{2}) to H^{∗}(2^{4}) is exactly the intersection L ∩ H^{∗}(2^{4})^{3×A_{4}}.

It also turns out that for Bockstein reasons the radical in H^{∗}(J_{2}; 𝔽_{2}) is the entire 𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11}).

In particular, the cohomology ring of J_{2} with 𝔽_{2}-coefficients is given as an extension

𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11})⟶H^{∗}(J_{2})⟶

𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}](1, c_{3}, R_{6}, X_{8}, Y_{8}, X_{9}, Y_{9}, c_{3}R_{6}, X_{12}, Y_{12}, X_{13}, Y_{13}, l_{15}, V_{18}, c_{3}V_{18})

⊕ 𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}]h_{14} ⊕ 𝔽_{2}[d_{3}, d_{8}, d_{12}]h_{15}.

Here, Sq^{1}(c_{3}) = d_{2}^{2}, Sq^{1}(d_{2}) = d_{3}, Sq^{1}(X_{2i}) = X_{2i+1}, Sq^{1}(Y_{2i}) = Y_{2i+1}, and there are Bocksteins β_{4}(k_{5}) = R_{6}, β_{8}(a_{7}) = d_{8}, β_{8}(a_{11}) = d_{12}. Some higher Bockstein of V_{18} is equal to d_{8}a_{11} + d_{12}a_{7} but we don’t know exactly which one.

Remark: Maginnis had already shown in [Mag2] that the homomorphism Φ: W → J_{2} above induces isomorphisms in mod(2) cohomology.

The basic differences in the treatments of J_{2} and J_{3} are necessitated by the fact that the normalizer in J_{3} of the 2^{4} is no longer the normalizer of 2^{4} in 2^{2+4}: (3 × S_{3}) but rather is the maximal subgroup 2^{4}: (3 × A_{5}). Consequently, the restriction image of H^{∗}(J_{3}; 𝔽_{2}) in H^{∗}(2^{4}) is contained in the restriction image of H^{∗}(J_{2}; 𝔽_{2}), and it follows that it is

res^{∗}(H^{∗}(J_{3}; 𝔽_{2})) = H^{∗}(2^{4})^{3×A_{5}} ∩ L.

Again, a check of Bocksteins shows that 𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11}) is the radical for H^{∗}(J_{3}; 𝔽_{2}).

Consequently, the resulting cohomology groups of J_{3} are given by the exact sequence

0⟶𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11})⟶H^{∗}(J_{3})⟶M⟶0

where M is Cohen-Macaulay, free as a module on 237 generators over the polynomial algebra

𝔽_{2}[X_{10}, X_{12}, X_{15}, X_{24}]

and has Poincaré series

\frac{p(x)}{(1 − x^{10})(1 − x^{12})(1 − x^{15})(1 − x^{24})}

with p(x) given as

p(x) = 1 + x^{6} + 3x^{8} + 2x^{9} + 2x^{11} + 3x^{12} + 2x^{13} + 5x^{14} + 5x^{15} + 5x^{16}
+ 6x^{17} + 4x^{18} + 4x^{19} + 8x^{20} + 7x^{21} + 7x^{22} + 9x^{23} + 8x^{24}
+ 8x^{25} + 9x^{26} + 8x^{27} + 10x^{28} + 11x^{29} + 8x^{30} + 9x^{31} + 8x^{32}
+ 8x^{33} + 9x^{34} + 7x^{35} + 7x^{36} + 9x^{37} + 4x^{38} + 4x^{39} + 6x^{40}
+ 5x^{41} + 5x^{42} + 5x^{43} + 2x^{44} + 4x^{45} + 2x^{46} + 2x^{48} + 3x^{49}
+ x^{51} + x^{57}

The details of these calculations are given in §7.

We now briefly discuss the relation between this calculation for H^{∗}(J_{3}) and the internal structure of J_{3}. Ryba, Smith, and Yoshiara, [RSY], constructed a two dimensional sporadic geometry based on the triple of 2-locals above. This geometry satisfies the condition that the fixed point set of each nontrivial 2-subgroup of J_{3} is contractible, and thus the 2-primary component of the cohomology of J_{3}, with coefficients in any ℤ(J_{3})-module, can be described in terms of the cohomology of the isotropy groups, as in Webb [W1,W2]. Instead of giving Webb’s alternating sum formula or exact sequence of cohomologies, we give an isomorphism between direct sums of cohomologies, with the cohomology of J_{3} and of the isotropy groups of odd dimensional simplexes appearing on one side, and the cohomology of the isotropy groups of even dimensional simplexes appearing on the other side. This result can be described topologically in terms of a homotopy colimit of the diagram of the classifying spaces of the istropy groups, as in Dwyer [D] (take the disjoint union of the simplexes crossed with the classifying spaces of their isotropy groups, and join together along the boundaries of the simplexes using maps induced by the inclusion of subgroups).

._{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}•  • • 2_{−}^{1+4}: A_{5}  2^{2+4}: (3 × S_{3}) 2^{4}: (3 × A_{5})^{S_{J}:3} _{2^{2+4}:3^{2}}_{2^{4}:(3×2^{2})}_{2^{4}:(3×2^{2})} The Ryba, Smith, Yoshiara sporadic geometry for J_{3}

On the other hand, since the isotropy group of the edge between 2_{−}^{1+4}: A_{5} and 2^{4}: (3 × A_{5}) is the same as that for the 2-simplex, the cohomology of the group 2^{4}: (3 × 2^{2}) appears on both sides of the isomorphism between direct sums of cohomologies, and can be canceled. So from the geometry we get the following formula for the 2-primary component of cohomology with coefficients in any ℤ(J_{3})-module M:

\begin{matrix}H^{∗}(J_{3}; M)_{2}⊕ & H^{∗}(S_{J}: 3; M)_{2} ⊕ H^{∗}(2^{2+4}: 3^{2}; M) \\ = H^{∗}(2^{1+4}: A_{5}; M)_{2} ⊕ H^{∗}(2^{2+4}: (3 × S_{3}); M)_{2} ⊕ H^{∗}(2^{4}: (3 × A_{5}); M)_{2} \end{matrix}

The one dimensional sporadic geometry for J_{2} only yields the cohomology of J_{2} with trivial coefficients, so, in §8, we also include a description of the Quillen complex Δ for J_{2}, with quotient Δ/J_{2} which is a three dimensional complex with 9 vertices, 23 edges, 25 faces, and 10 quads. The quotient is contractible, being a cone on a two dimensional subcomplex. See §8 for a description of the faces and quads, but the following picture illustrates the 1-skeleton of the subcomplex, where A^{i}B^{j} denotes an elementary abelian 2-subgroup with i central involutions and j noncentral involutions. Note that the cone point A^{3}B^{12} is not drawn here, so that the 1-skeleton of Δ/J_{2} is the 1-skeleton of the cone on the picture.

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 •  • •  •  •  •  • A^{3}  A^{1}  A^{1}B^{6} A^{3}B^{4}  A^{1}B^{2}  B^{1}B_{II}^{3}  B_{I}^{3} The 1-skeleton of a subcomplex of Δ/J_{2}

In particular, (8.2) gives the following description of H^{∗}(J_{2}; M)_{2} for any ℤ(J_{2})-module M:

\begin{aligned}H^{n}(J_{2}; M)_{2} = & H^{n}(2_{−}^{1+4}: A_{5}; M)_{2} + H^{n}(2^{2+4}: (3 × S_{3}); M)_{2} \\ & − H^{n}(S_{J}: 3; M)_{2} + H^{n}(A_{4} × A_{5}; M)_{2} − H^{n}(A_{4} × A_{4}; M)_{2}. \end{aligned}

In the process of proving Theorem A we have to make a closer analysis of the cohomology ring H^{∗}(2^{2+4}) than was done in [Mag] or [AM1]. In particular we show that H^{∗}(2^{2+4}) is detected by restriction to its two 2^{4}’s together with its three copies of Q_{8} × 2. Also, we show that 2_{−}^{1+4} contains exactly five copies of Q_{8} × 2 ⊂ 2_{−}^{1+4} which detect its cohomology.

This gives all seven copies of Q_{8} × 2 ⊂ S_{J}. They decompose into three conjugacy classes in S_{J} while they fuse into a single conjugacy class in W and hence in any homomorphic image of W.

§1. The detailed structure of 2^{2+4}

The Sylow 2-subgroup of J_{2} is described as follows: there is an index two subgroup isomorphic to 2^{2+4}, and S_{J} is a split extension 2^{2+4}: 2 where the extending 2 acts to interchange the two 2^{4}’s and acts non-trivially on the central 2^{2}. More precisely, regard 2^{2+4} as the group of upper triangular matrices in GL_{3}(𝔽_{4}), 2^{2+4} = {μ(α, β, χ)|α, β, γ ∈ 𝔽_{4}} where

μ(α, β, χ) = ( \begin{matrix}1 & α & β \\ 0 & 1 & γ \\ 0 & 0 & 1 \end{matrix} ) .

For later reference we name the generators for the groups. Let ξ be a primitive third root of unity in 𝔽_{4} and let

(1.1)  \begin{matrix}x = μ(1, 0, 0), & y = μ(ξ, 0, 0), \\ z = μ(0, 1, 0), & t = μ(0, ξ, 0), \\ e = μ(0, 0, 1), & f = μ(0, 0, ξ). \end{matrix}

There are exactly two 2^{4}’s in 2^{2+4}, the subgroup 2_{I}^{4} = ⟨x, y, z, t⟩, and the subgroup 2_{II}^{4} = ⟨e, f, z, t⟩, and both are normal in 2^{2+4}. Also note that the center Z(2^{2+4}) = 2^{2} = ⟨z, t⟩. We have the following.

Lemma 1.2: Out(2^{2+4}) is an extension of the form:

Out(2^{2+4}) ≅ 2^{4}: (3^{2}: 2^{2})

where N_{Out(2^{2+4})}(3^{2}) = 3^{2}: 2^{2} = S_{3} × S_{3}. The subgroup 2^{4}: 3^{2} ≅ A_{4} × A_{4}.

Proof: There is an automorphism ρ of order two,

(1.3)  ρ: ( \begin{matrix}1 & α & β \\ 0 & 1 & γ \\ 0 & 0 & 1 \end{matrix} ) ↔ ( \begin{matrix}1 & γ & β \\ 0 & 1 & α \\ 0 & 0 & 1 \end{matrix} )^{−1} ,

and thus, given an arbitrary element of Aut(2^{2+4}), we can assume, after perhaps multiplying by ρ, that it takes 2_{II}^{4} to itself. It follows that restriction to 2_{II}^{4} maps an index two subgroup of Aut(2^{2+4}) which we denote F (2_{II}^{4}) into GL_{4}(2). But Z(2^{2+4}) = 2^{2} is a subgroup of 2_{II}^{4}, and the flag Z(2^{2+4}) ⊂ 2_{II}^{4} must be preserved by any element in F (2_{II}^{4}). Consequently, the action of F (2_{II}^{4}) on 2_{II}^{4} defines a homomorphism Γ : F (2_{II}^{4}) ⟶ P where P is the the parabolic subgroup of L_{4}(2) of the form

P = ( \begin{matrix}L_{2}(2) & ∗ \\ 0 & L_{2}(2) \end{matrix} ) .

The image under Γ of the inner automorphism group of 2^{2+4} in F (2_{II}^{4}) is just the image of the subgroup generated by the conjugations by the elements of J = ⟨x, y⟩. The image has the form

⟨( \begin{matrix}I & I \\ 0 & I \end{matrix} ) , ( \begin{matrix}I & Ξ \\ 0 & I \end{matrix} )⟩

where Ξ = ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ). Of course this assumes a choice of basis \{z, t, e, f\} for 2_{II}^{4}.

The group consisting of matrices of the form ( \begin{matrix}I & ∗ \\ 0 & I \end{matrix} ) thus centralizes the image of

the inner automorphisms and is easily seen to lift to automorphisms fixing the subgroup J. Also, L_{2}(2) ≅ S_{3}, and a subgroup 3^{2} ⊂ Aut(2^{2+4}) is generated by the automorphisms T and τ, explicitly constructed as conjugation by the respective matrices

(1.4)  ( \begin{matrix}1 & 0 & 0 \\ 0 & ξ & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ξ \end{matrix} ) .

An element of order two acting to invert, simultaneously, both of the generators of the 3^{2} above is the galois automorphism σ: 𝔽_{4}⟶𝔽_{4} acting simultaneously on all the coefficients in our matrix representation of 2^{2+4}. Finally, the constraint that the image of F (2_{II}^{4}) in L_{4}(2) normalize the image of Inn(2^{2+4}) is directly seen to imply that the element

( \begin{matrix}V & 0 \\ 0 & I \end{matrix} ) is not in Γ(F (2_{II}^{4})), where V = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). It follows that we have constructed

the entire image, and it is an index two subgroup of the (2, 2)-parabolic above. Moreover, the quotient by the inner automorphisms gives an extension of the form 2^{2}: (3^{2}: 2) for the image of the corresponding index two subgroup of Out(2^{2+4}).

On the other hand, the kernel of this map consists of central automorphisms on J, which, by reflecting the argument above, are seen to give a single extra copy of 2^{2} in Out(2^{2+4}). From this the result follows directly.

Corollary 1.5: 2^{2+4} has 15 index two subgroups, 6 copies of 2^{2} ≀ 2 and 9 copies of the split product 4^{2}: 2 where the element of order two, n, acts on the two generators of the 4^{2}, (s, r), by s^{n} = s^{−1}r^{2}, r^{n} = r^{−1}.

Proof: The index two subgroups are in one to one correspondence with the non-trivial homomorphisms G/G′ = 2^{4}⟶2 for G = 2^{2+4}. The action of the 3^{2} ⊂ Out(2^{2+4}) decomposes the non-zero elements of Hom(2^{4}, 2) = 2^{4} into three orbits, two containing three elements each, and one with 9 elements, while the automorphism which exchanges the two 2^{4}’s also exchanges the two orbits of three elements. It follows that there are 6 isomorphic subgroups corresponding to the first orbit and 9 corresponding to the second.

A representative for the first orbit is the subgroup described by constraining α in (1.3) to be either 0 or 1. It contains 2_{II}^{4} and is consequently given as an extension of the form 2^{4}: 2. It is easily seen that this group is a copy of 2^{2} ≀ 2. For the second type a set of generators is given by the elements

(1.6)  s = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , r = ( \begin{matrix}1 & ξ & 0 \\ 0 & 1 & ξ \\ 0 & 0 & 1 \end{matrix} ) , n = ( \begin{matrix}1 & ξ & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) (= y).

Morover the first two span a 4^{2} and the third acts in the way specified.

Now we determine the set of index 4 subgroups of 2^{2+4}, paying particular attention to the copies of Q_{8} × 2 ⊂ 2^{2+4}. Lemma 1.7: The group 2^{2+4} contains exactly three copies of Q_{8} × 2, and each is normal. Proof: The group 4^{2}: 2 ⊂ 2^{2+4} described in (1.6) has subgroup structure given as follows. Denote the homomorphism f: (4^{2}: 2)⟶2 sending s to ϵ_{1}, r to ϵ_{2}, n to ϵ_{3} by the triple (ϵ_{1}, ϵ_{2}, ϵ_{3}) where ϵ_{i} = 0 or 1, then we have the following table for the kernels of the seven homomorphisms:

(1.8)

homomorphism  kernel  name
(1, 0, 0)  ⟨s^{2}, r, n⟩  D_{8} × 2
(0, 1, 0)  ⟨s, r^{2}, n⟩ (4 × 2):_{A} 2
(0, 0, 1) ⟨s, r⟩ 4^{2}
(1, 1, 0)  ⟨s^{2}, sr, n⟩ (4 × 2):_{B} 2
(1, 0, 1) ⟨s^{2}, r, sn⟩ Q_{8} × 2
(0, 1, 1)  ⟨s, rn, r^{2}⟩ (4 × 2):_{A} 2
(1, 1, 1)  ⟨s^{2}, rs, sn⟩ (4 × 2):_{B} 2

where A denotes the action on 4 × 2 = ⟨b, c⟩ given by aba = b^{−1}c, and aca = c, while B denotes the action given by aba = ybc, aca = c.

The interesting subgroup in (1.8) is the Q_{8} × 2. It is normal in 2^{2+4}, and from the 3^{2}-action, there are exactly three of them that arise in this way, each lying in three of the groups 4^{2}: 2.

The subgroup structure of 2^{2} ≀ 2 is given as follows. There is one copy of 2^{4}, three copies of 2 × D_{8}, and three copies of the twisted extension

2^{2}: 4 ≅ ⟨a, b, w | ⟨a, b⟩ = 2^{2}, a^{w} = b, b^{w} = a⟩

so 2^{2+4} contains only the three copies of Q_{8} × 2 above. Remark 1.9: Each of the three copies of Q_{8} × 2 ⊂ 2^{2+4} are determined uniquely by the involution in its commutator subgroup (either z, t or tz in the notation of (1.1).

Finally, we point out the following useful result. Lemma 1.10: Any two of the Q_{8} × 2 ⊂ 2^{2+4} generate the entire group 2^{2+4}. Proof: Each index 2 subgroup of 2^{2+4} contains at most one copy of Q_{8} × 2.

§2. The construction of 2^{2+4}: (3 × S_{3})

Lemma 2.1: There is, up to isomorphism, one and only one group which has the form a split extension 2^{2+4}: (3 × S_{3}) with the extending element of order two acting non-trivially on the center and exchanging 2_{I}^{4}, 2_{II}^{4} and which has the property that the induced map 3^{2}: 2⟶Aut(2^{2+4}) is injective.

Proof: From Lemma 1.2 we see that 3^{2} = Syl_{3}(Aut(2^{2+4})). Hence, up to isomorphism we can assume that the 3^{2} acts on 2^{2+4} as it does in GL_{3}(4) in the upper triangular matrices. Also, using the automorphism ρ in (1.3), we see that ρ extends to the upper triangulars by exchanging

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ξ \end{matrix} ) ↔ ( \begin{matrix}ξ^{−1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,  ( \begin{matrix}1 & 0 & 0 \\ 0 & ξ & 0 \\ 0 & 0 & 1 \end{matrix} ) ↔ ( \begin{matrix}1 & 0 & 0 \\ 0 & ξ^{−1} & 0 \\ 0 & 0 & 1 \end{matrix} ) .

This is not quite the automorphism we need however. What we use is the composition of the Galois automorphism, σ, with ρ, which has order 2, normalizes the 3^{2} in Aut(2^{2+4}) and acts, as desired, non-trivially on the center while it exchanges the two 2^{4}’s. Since the normalizer of 3^{2} in Aut(2^{2+4}) is 3^{2}: 2^{2} from 1.2, we see that A = ρσ is the unique element which satisfies the constraints on the action. We still have to verify that this is the unique extension having this action. However, the center of 2^{2+4}: 3^{2} is trivial, thus standard results on extensions show that the split extension above is the unique extension of this kind. So 2.1 follows.

Next we construct this group explicitly. To begin, from (1.1), we can write down the relations for 2^{2+4}. They are

(2.2)  [x, e] = z, [x, f] = t, [y, e] = t, and [y, f] = tz.

and the three Q_{8} × 2’s are given as

(2.3)  \begin{aligned}(Q_{8} × 2)_{z} & = ⟨xe, xyf, t⟩, \\ (Q_{8} × 2)_{t} & = ⟨xf, ye, z⟩, \\ (Q_{8} × 2)_{tz} & = ⟨yf, xye, t⟩. \end{aligned}

The proof of 2.1 now shows that S_{J} is given by adding a single generator, A, of order 2 to the list above with action given by

(2.4)  x^{A} = e, y^{A} = ef, z^{A} = z, t^{A} = tz.

In this notation we see that the center of S_{J}, Z(S_{J}) = ⟨z⟩. Moreover, the central quotient

(2.5)  S_{J}/⟨z⟩ ≅ UT_{4}(2),

(see e.g. [Mag], this is the group of upper triangular matrices in L_{4}(2)), and UT_{4}(2) has the form of an extension 2^{4+2} with the 2^{4} = ⟨Ā, \overline{xe}, \overline{fxy}, t̄⟩ and the extending 2^{2} given as ⟨x̄, ȳ⟩†. Then, the set of 2^{3} ⊂ UT_{4}(2) are given up to conjugacy as subgroups of the 2^{4} or as one of the four groups

(2.6) Group ⟨x̄, ȳ, t̄⟩ ⟨x̄, \overline{xe}, t̄⟩ ⟨ȳ, \overline{ef}, t̄⟩ ⟨\overline{xy}, f̄, t̄⟩

Lift  2^{4}  2 × D_{8}  2 × D_{8}  2 × D_{8}

where we have included the inverse image in S_{J} of each of these groups in the second line of the table above.

Also, we have an explicit generator, T, for the 3 which normalizes S_{J} in J_{2}, A ↦ A, x⟶y⟶xy, f⟶e⟶ef, and T fixes z, t,

and a generator, τ, for a remaining 3 in N_{J_{2}}(2^{2+4}):

x and y are fixed, e ↦ f ↦ ef, z ↦ t ↦ zt.

Note the relations in Aut(2^{2+4})

AT = T A, Aτ = τ^{−1}T^{−1}A

which thus hold in the extension 2^{2+4}: (3^{2}: 2) as well.

A first connection between this and H^{∗}(G), for G a finite group of the type described in Theorem A, is given as follows:

Lemma 2.7: There is exactly one conjugacy class of 2^{4}’s in S_{J}. Consequently, the pair 2^{4} ⊂ S_{J} is weakly closed in G. It follows that the image of restriction

im (res^{∗}(H^{∗}(G)) ⊂ H^{∗}(2^{4})) = H^{∗}(2^{4})^{W_{G}(2^{4})} ∩ im (res^{∗}(H^{∗}(S_{J})) ⊂ H^{∗}(2^{4}))

where W_{G}(2^{4}) = N_{G}(2^{4})/2^{4} is the Weyl group of 2^{4} in G.

Proof: Observe that under the projection from S_{J} to S_{J}/ (Z(S_{J}) = ⟨z⟩) = UT_{4}(2),

[Mag], the inverse image of the subgroup 2^{4} is 2_{−}^{1+4} which has 2-rank 2. Consequently, the only possibilities for 2^{4}’s in S_{J} are subgroups which map to one of the 4 conjugacy classes of extremal 2^{3}’s in UT_{4}(2). But (2.6) shows that only subgroups conjugate to ⟨x, y, t⟩ have such inverse images, and since the normalizer of ⟨x, y, t⟩ has index two in UT_{4}(2) it follows that the only two 2^{4}’s in S_{J} are 2_{I}^{4} and 2_{II}^{4} in the subgroup 2^{2+4}.

The last assertion follows directly from the Cardenas-Kuhn theorem, [AM].

Remark 2.8: In particular, when G = J_{2} we see that W_{J_{2}}(2^{4}) = 2^{2}: 3^{2}, while for G = J_{3}, W_{J_{3}}(2^{4}) = GL_{2}(4).

It remains to consider the structure of Q_{8} ∗ D_{8} ⊂ S_{J} which we turn to next.

† We use a bar over a element of S_{J} to denote its image in UT_{4}(2).

§3. The subgroup Q_{8} ∗ D_{8} and the fusion of the Q_{8} × 2

There is a 2_{−}^{1+4} ≅ D_{8} ∗ Q_{8} = ⟨A, xe, fxy, t, z⟩ ◁ S_{J} with normalizer 2_{−}^{1+4}: A_{5} in J_{2}, where the action of A_{5} on the quotient 2^{4} is obtained from the permutation action on 2^{5} by restricting to the codimension one subspace of even vectors. Moreover, the intersection of this 2_{−}^{1+4} with 2^{2+4} is ⟨xe, fxy, t, z⟩ ≅ (Q_{8} × 2)_{z}, and is the commutator subgroup of S_{J}.

We give a smaller presentation for D_{8} ∗ Q_{8} which is more convenient for studying the subgroup structure:

(3.1)  D_{8} ∗ Q_{8} = {a, b, c, d | ⟨a, b⟩ = D_{8}, ⟨c, d⟩ = Q_{8}, D_{8} ∩ Q_{8} = ⟨z⟩ = 2}.

These relate to the generators given above by letting a = A, b = t, c = xe, and d = fxy. From this we see that there are exactly five conjugacy classes of non-central elements of order two given as {a, za}, {b, zb}, {abc, abc^{−1}}, {abd, abd^{−1}}, {abcd, ab(cd)^{−1}}, which are permuted transitively under the A_{5} action. Hence they all have isomorphic centralizers, copies of Q_{8} × 2. Conversely, any Q_{8} × 2 ⊂ 2_{−}^{1+4} centralizes a 2^{2} and hence a non-central 2 in 2_{−}^{1+4}. It follows that D_{8} ∗ Q_{8} contains exactly 5 copies of Q_{8} × 2, and it can be easily verified that the remaining ten index two subgroups all have the form 4 ∗ Q_{8} ≅ 4 ∗ D_{8}.

Lemma 3.2: There are three conjugacy classes of subgroups Q_{8} × 2 in S_{J}, and they fuse into a single conjugacy class in any G which is a homomorphic image of the amalgamated product W described in Theorem A. Moreover, the normalizer of Q_{8}×2 in W is isomorphic to S_{J}: 3 and has the form of an extension Q_{8} × 2 · (2^{3}: 3).

Proof: Since Syl_{2}(A_{5}) is the Klein group K = ⟨(1, 3)(2, 4), (1, 2)(3, 4)⟩ ⊂ A_{4} it follows that the five copies of Q_{8} × 2 ⊂ 2_{−}^{1+4} break up into two conjugacy classes in S_{J}, one containing 4 groups, and the other containing the normal Q_{8} × 2 given as 2_{−}^{1+4} ∩ 2^{2+4}. Likewise, the three Q_{8} × 2 in 2^{2+4} break up into 2 conjugacy classes, the first given as the intersection above and the remaining two given in the second conjugacy class since the action of A on 2^{2+4} does not leave the center fixed.

It remains to show that these are the only copies of Q_{8} × 2 ⊂ S_{J}, and to verify that they all fuse in W.

For any Q_{8} × 2 ⊂ S_{J} we have that its image in S_{J}/⟨z⟩ ≅ UT_{4}(2) is a 2^{3}, a Q_{8} × 2 or a copy of Q_{8}. However, it cannot be a Q_{8} × 2 since UT_{4}(2) does not contain any copies of this group. Thus it must be a Q_{8} or a 2^{3}.

There are, as shown in [M], exactly two copies of Q_{8} in UT_{4}(2), ⟨\overline{xf}, \overline{ye}⟩ and ⟨\overline{xef}, \overline{yf}⟩ both of which lift to copies of 2 × Q_{8} contained in 2^{2+4}. For the 2^{3} ⊂ UT_{4}(2) which are not contained in the 2^{4} we have seen in (2.6) that they lift to either 2^{4}’s or 2 × D_{8}’s. On the other hand, the lift of the 2^{4} is precisely 2_{−}^{1+4}. Consequently, the only subgroups Q_{8} × 2 ⊂ S_{J} are the seven groups that we have already enumerated.

The fusion of the two conjugacy classes in 2^{2+4} is accomplished by the element of order three in N_{J_{2}}(2^{2+4}) which does not fix the center. The fusion of the two conjugacy classes in 2_{−}^{1+4} is accomplished directly since the action of the normalizing A_{5} on the five subgroups of this form is the usual permutation action of A_{5} on the set of five elements. In particular, any element of order five acts transitively.

Next, since the intersection of 2_{−}^{1+4} and 2^{2+4} is a normal Q_{8} × 2, and the fact that N_{A_{5}}(Q_{8} × 2) = A_{4} the remaining statement follows.

As a consequence we have the following result.

Corollary 3.3: There is exactly one copy of 2^{2+4} ⊂ S_{J} and also only one copy of the subgroup 2_{−}^{1+4}.

Proof: First consider a copy of 2^{2+4} ⊂ S_{J}. It must be normal, hence it contains all the Q_{8} × 2 in a conjugacy class if it contains any. Then 3.2 and 2.10 imply the result.

The subgroup 2_{−}^{1+4} must contain five of the seven Q_{8} × 2’s. However, each has a distinguished element, the only non-zero element which is a square, and from the form of 2_{−}^{1+4} the distinguished element must be the same in all five copies. This forces the new 2_{−}^{1+4} to contain the same five Q_{8} ×2’s as did the original, but as above these five subgroups span the entire group.

§4. The cohomology ring of 2^{2+4} and the restriction to H^{∗}(2_{I}^{4})

There are three steps remaining in determining the cohomology of J_{2}, J_{3}. In this section we show that the three Q_{8} × 2 and the 2^{4} detect H^{∗}(S_{J}). So 2_{I}^{4} together with (Q_{8} × 2)_{z} ⊂ S_{J} detect H^{∗}(G) for any finite G with Sylow subgroup S_{J} given as the homomorphic image of W. Then we evaluate the image of restriction in H^{∗}(2_{I}^{4}). The third step will be accomplished in the next section where we determine the radical in H^{∗}(J_{i}) by first evaluating H^{∗}(2^{2+4})^{3×S_{3}}, using the detection result above to show that the radical of H^{∗}(J_{i}) is contained in the radical of H^{∗}(2^{2+4}), and finally showing that the radical of H^{∗}(J_{i}) must be at least as large as the invariants in the radical of H^{∗}(2^{2+4}).

From [AM1] – see also the alternate calculation in [Mag] – we have that H^{∗}(2^{2+4}) has generators x, y, e, f, L_{3}, M_{3}, v_{4}, w_{4}, R_{6} and expands as 𝔽_{2}[v_{4}, w_{4}] tensored with

(4.1)  (𝔽_{2}[x, y] ⊕ 𝔽_{2}[e, f](e) ⊕ 𝔽_{2}[f](f))(1, L, M, LM) ⊕ (ye, yf, y^{2}e, y^{2}f, yeL, R)

with L, M both in the image from H^{∗}(S_{J}). Here R_{6} restricts to the image of LM in H^{∗}(2_{I}^{4}) and to zero in H^{∗}(2_{II}^{4}), and the radical is given as

(4.2)  𝔽_{2}[v_{4}, w_{4}](ye, yf, y^{2}e, y^{2}f, yeL_{3}).

Lemma 4.3: The two 2^{4}’s and the three copies of Q_{8} × 2 in 2^{2+4} detect the mod(2) cohomology of 2^{2+4} under restriction.

Proof: From [AM1] we have that the radical is exactly the kernel of restriction to H^{∗}(2_{I}^{4}) ⊕ H^{∗}(2_{II}^{4}),

Ker (res^{∗}: H^{∗}(2^{2+4})⟶H^{∗}(2_{I}^{4}) ⊕ H^{∗}(2_{II}^{4})) = 𝔽_{2}[v_{4}, w_{4}](ye, yf, y^{2}e, y^{2}f, yeL_{3}).

Moreover, in H^{∗}(Q_{8} × 2) = 𝔽_{2}[γ_{4}, t_{1}](1, a, b, a^{2}, b^{2}, a^{2}b) a computer generated calculation using MAGMA, (see §9), gives the following restriction table where a is dual to the first generator in the corresponding group of (2.3) while b is dual to the second and t is dual to the third:

(4.4)  \begin{matrix}element & (Q_{8} × 2)_{z} & (Q_{8} × 2)_{t} & (Q_{8} × 2)_{zt} \\ x & a + b & a & b \\ y & b & b & a + b \\ e & a & b & b \\ f & b & a & a \\ L_{3} & b^{2}t + bt^{2} & (a + b)^{2}t + (a + b)t^{2} & b^{2}t + bt^{2} \\ v_{4} & t^{4} + a^{2}t^{2} & γ_{4} + a^{2}t^{2} & t^{4} + γ_{4} \\ w_{4} & γ_{4} & γ_{4} + t^{4} & γ_{4} \end{matrix}

Consequently we have the following restriction images for the generators of the radical:

(4.5)  \begin{matrix}element & (Q_{8} × 2)_{z} & (Q_{8} × 2)_{t} & (Q_{8} × 2)_{zt} \\ ye & ab & b^{2} & a^{2} \\ yf & b^{2} & ab & b^{2} \\ y^{2}e & a^{2}b & 0 & a^{2}b \\ y^{2}f & 0 & a^{2}b & a^{2}b \\ yeL_{3} & a^{2}bt^{2} & a^{2}bt^{2} & a^{2}bt^{2} \\ v_{4} & t^{4} + a^{2}t^{2} & γ_{4} + a^{2}t^{2} & t^{4} + γ_{4} \\ w_{4} & γ_{4} & γ_{4} + t^{4} & γ_{4} \end{matrix}

and the image clearly has the form 𝔽_{2}[v_{4}, w_{4}](ye, yf, y^{2}e, y^{2}f, yeL_{3}) as required. Corollary 4.6: The restriction H^{∗}(G)⟶H^{∗}(2^{2+4}) is injective for any G as above. Proof: Rusin [Ru] shows that three of the Q_{8} × 2 together with a number of 4 × 2 detect H^{∗}(2_{−}^{1+4}). But, since 2 × 2 ⊂ 4 × 2, it follows that each 4 × 2 is contained in one of the Q_{8} × 2’s. Thus the five Q_{8} × 2 ⊂ 2_{−}^{1+4} detect H^{∗}(2_{−}^{1+4}).

We have seen that the two 2^{4} ⊂ 2^{2+4} together with the three Q_{8} × 2 ⊂ 2^{2+4} detect the cohomology of 2^{2+4}. But the result of [Mag] quoted in the introduction shows that the two subgroups 2_{−}^{1+4} and 2^{2+4} detect H^{∗}(S_{J}). Consequently, any one of the Q_{8} × 2 ⊂ 2^{2+4} together with one of the 2^{4}’s detect H^{∗}(W).

We now conclude this section by recalling from [AM1] the exact structure of the image of the restriction from H^{∗}(2^{2+4}) to H^{∗}(2_{I}^{4}). Lemma 4.7: The image of restriction from H^{∗}(2^{2+4}) to H^{∗}(2_{I}^{4}) is

𝔽_{2}[x, y, v_{4}, w_{4}](1, L, M, LM) = 𝔽_{2}[x, y, t, z]^{2^{2}}

where

\begin{aligned}v_{4} & = t^{4} + t^{2}d_{2} + td_{3} \\ w_{4} & = (t + z)^{4} + (t + z)^{2}d_{2} + (t + z)d_{3} \\ L & = x^{2}t + t^{2}x + y^{2}(z + t) + y(z + t)^{2} \\ M & = y^{2}t + yt^{2} + (x + y)^{2}(z + t) + (x + y)(z + t)^{2} \end{aligned}

and d_{2} = x^{2} + xy + y^{2}, d_{3} = xy(x + y). Also,

L^{2} = x^{2}v + y^{2}w + d_{3}L
M^{2} = (x + y)^{2}w + y^{2}v + d_{3}M
.

§5. The invariants H^{∗}(2^{4})^{2^{2}:3^{2}}

In §2 we constructed an explicit model for the group 2^{2+4}: (3 × S_{3}). In this group, the Weyl group of 2_{I}^{4} has the form 2^{2}: 3^{2} so the normalizer of 2_{I}^{4} has the form

2^{4}: (2^{2}: 3^{2}) = 2^{2+4}: 3^{2}

and the generators, T and τ, for the action of 3^{2} are given by

(5.1)  \begin{aligned}T (x) & = y \\ T (y) & = x + y \\ T (t) & = t \\ T (z) & = z \end{aligned}  \begin{aligned}τ(x) & = x \\ τ(y) & = y \\ τ(t) & = z \\ τ(z) & = z + t \end{aligned}

Lemma 5.2: The ring of invariants under the action of 2^{2}: 3^{2} is

𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}]((1, c_{3}, c_{12}, c_{3}c_{12})(1, R_{6}), X_{8}, Y_{8}, X_{9}, Y_{9}, X_{12}, Y_{12}, X_{13}, Y_{13}).

This is a free module over 𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}] on 16 generators where d_{8}, d_{12} are the Dickson invariants, c_{3} = x^{3} + y^{3} + x^{2}y, c_{12} = v^{3} + w^{3} + v^{2}w, R = L^{2} + M^{2} + LM, and

\begin{aligned}X_{8} & = w(yL + xM) + (v + w)((x + y)L + yM) \\ Y_{8} & = (v + w)(yL + xM) + w((x + y)L + yM) \\ X_{9} & = w(y^{2}L + x^{2}M) + (v + w)((x + y)^{2}L + y^{2}M) \\ Y_{9} & = (v + w)(y^{2}L + x^{2}M) + w((x + y)^{2}L + y^{2}M) \\ X_{12} & = w^{2}(yL + xM) + (v + w)^{2}((x + y)L + yM) \\ Y_{12} & = (v + w)^{2}(yL + xM) + w^{2}((x + y)L + yM) \\ X_{13} & = w^{2}(y^{2}L + x^{2}M) + (v + w)^{2}((x + y)^{2}L + y^{2}M) \\ Y_{13} & = (v + w)^{2}(y^{2}L + x^{2}M) + w^{2}((x + y)^{2}L + y^{2}M). \end{aligned}

Proof: Part of 4.7 is the assertion that 𝔽_{2}[x, y, v_{4}, w_{4}](1, L, M, LM) is the invariant subring H^{∗}(2^{4})^{2^{2}}. We start by determining the invariants in this ring under the action of T. Note that v, w are fixed by T, while L ↦ M ↦ L + M. It follows that R is fixed, and we can write 𝔽_{2}[x, y] = 𝔽_{2}[d_{2}, d_{3}](1, c_{3}, x, x^{2}, y, y^{2}). Thus, it is directly seen that the invariants under T are given as

𝔽_{2}[v, w, d_{2}, d_{3}](1, c_{3}, R, c_{3}R, yL + xM, y^{2}L + x^{2}M, (x + y)L + yM, (x + y)^{2}L + y^{2}M).

Now we apply τ to this ring. Note that τ acts as

\begin{matrix}M ⟶L⟶L + M \\ w⟶v⟶v + w \end{matrix}

and, as directly above, we write

(5.3)  \begin{aligned}𝔽_{2}[v, w] & = 𝔽_{2}[v^{2} + vw + w^{2}, v^{2}w + vw^{2}](1, c_{12}) \\ & ⊕ 𝔽_{2}[v^{2} + vw + w^{2}, v^{2}w + vw^{2}](v, w, v^{2}, w^{2}). \end{aligned}

The first term in (5.3) gives rise to the part of the invariant subring of the form

𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}](1, c_{3})(1, c_{12})(1, R_{6}),

and the second part gives rise to the remainder exactly as above.

Corollary 5.4: The Poincaré series of the ring of invariants 𝔽_{2}[x_{1}, …, x_{4}]^{W_{J_{2}} (2^{4})} is given as

\frac{1 + t^{3} + t^{6} + 2t^{8} + 3t^{9} + 3t^{12} + 2t^{13} + t^{15} + t^{18} + t^{21}}{(1 − t^{2})(1 − t^{3})(1 − t^{8})(1 − t^{12})} .

It is useful to give the structure of the Sq^{1}-homology of this ring of invariants. We have Sq^{1}(c_{3}) = d_{2}^{2}, Sq^{1}(X_{2j}) = X_{2j+1}, Sq^{1}(Y_{2j}) = Y_{2j+1}, and Sq^{1}(d_{2}) = d_{3}. From this it is direct to see that

(5.5)  H_{∗}(𝔽_{2}[x_{1}, …, x_{4}]^{W_{J_{2}} (2^{4})}; Sq^{1}) = 𝔽_{2}[d_{8}, d_{12}](1, R_{6}, c_{12}, c_{12}R_{6}).

We now determine the intersection of the ring of invariants in 5.2 with H^{∗}(S_{J}). We begin by describing the image of H^{∗}(S_{J}) in H^{∗}(2^{4})^{2^{2}}.

Proposition 5.6:

\begin{aligned}Im(res_{2^{4}}^{S_{J}}) = & 𝔽_{2}[x, y, v_{4}, w_{8}](1, L, M, LM, N_{1}, LN_{1}, MN_{2}, W) \\ & ⊕ 𝔽_{2}[y, v_{4}, w_{8}](N_{2}, LN_{2}, MN_{2}) \end{aligned}

where w_{8} = w_{4}^{2} + w_{4}v_{4}, N_{1} = xw_{4}, N_{2} = yw_{4}, and W = res_{2^{4}}^{S_{J}} · T r_{2^{2+4}}^{S_{J}}(w_{4}R_{6}).

Proof: This follows from the additive description of H^{∗}(S_{J}; 𝔽_{2}) given in [Mag] along with the trivial restrictions res(x) = 0, res(α) = 0, res(β) = 0. The independence of the remaining restrictions follows from a filtration of 𝔽_{2}[x, y, t, z] by powers of z, or equivalently from the description of the E_{∞}-page of the spectral sequence for the central extension in [Mag].

Proposition 5.7: The intersection of the image of restriction with the invariants

Im(res_{2^{4}}^{S_{J}}) ∩ H^{∗}(2^{4})^{W}

is given as

\begin{matrix}𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}] & (1, c_{3}, R_{6}, c_{3}R_{6}, X_{8}, Y_{8}, X_{9}, Y_{9}, X_{12}, Y_{12}, X_{13}, Y_{13}, c_{3}c_{12}, V_{18}, c_{3}V_{18}) \\ ⊕ 𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}]d_{2}c_{12} ⊕ 𝔽_{2}[d_{3}, d_{8}, d_{12}]d_{3}c_{12}. \end{matrix}

Proof: Clearly 𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}](1, c_{3}, R_{6}, c_{3}R_{6}, X_{8}, Y_{8}, X_{9}, Y_{9}, X_{12}, Y_{12}, X_{13}, Y_{13}) is in the intersection since d_{8} = w_{8} + v_{4}^{2} and d_{12} = v_{4}w_{8}. That the X_{i} and Y_{j} are in the intersection follows since x, y, L, and M are in the image of restriction. Moreover, N_{1} = xw_{4} and N_{2} = yw_{4} are in the image. Thus, for example,

X_{8} = res_{2^{4}}^{S_{J}} (M(N_{1} + N_{2}) + LN_{1} + v_{4}((x + y)L + yM)).

A similar argument shows that c_{3}c_{12}, c_{3}R_{6}c_{12}, d_{2}c_{12} and d_{3}c_{12} are in the intersection. For example

d_{2}c_{12} = res_{2^{4}}^{S_{J}} ((v_{4}w_{8} + v_{4}^{3})(x^{2} + xy + y^{2}) + w_{8}((x + y)N_{1} + yN_{2})).

Finally, R_{6}c_{12} is in the intersection, which is seen by defining an element in H^{18}(S_{J}; 𝔽_{2}) by

V_{18} = T r_{2^{2+4}}^{S_{J}}(c_{12}R_{6}).

Note that R_{6} is described in the remarks following (4.1).

In fact the only invariants that are not in the image of restriction are 𝔽_{2}[d_{8}, d_{12}]c_{12}. This can most easily be shown by restricting further to 2^{2} = ⟨t, z⟩. Here c_{12} is not invariant under the normalizerN_{J_{2}}(2^{2}) = 2^{2}: S_{3}; specifically, it is not invariant under the ℤ/2-action exchanging t and tz.

Comparing with the Sq^{1} cohomology of the invariant subring determined in (5.5) we have the the image of the Sq^{1} cohomology of H^{∗}(J_{2}) is contained in

(5.8)  𝔽_{2}[d_{8}, d_{12}](1, R_{6}, V_{18}).

Consequently, there must be an element in H^{5}(J_{2}) with some higher Bockstein equal to R_{6}, and there must be elements a_{7}, a_{11} with Bocksteins d_{8}, and d_{12}, respectively or there can be Bocksteins from these elements to elements in the radical. There is some Bockstein of V_{18} with image d_{12}a_{7} + d_{8}a_{11}, but we don’t know which one.

§6. The structure of the radical in the invariant subring H^{∗}(2^{2+4})^{3^{2}:2}

The action of 3^{2}: 2 on 2^{2+4} has been determined in §2. Also, the cohomology of 2^{2+4} has been described in §4. Note that the radical is always closed under the action of any subgroup of Aut(2^{2+4}). We now determine the subset of the radical that is invariant under the action of 3 × S_{3}. It turns out to be sufficiently small that it must all be in the image of H^{∗}(J_{2}) on comparing with (5.8).

Lemma 6.1: The set of invariants under the action of 3^{2}: 2 in the radical has the form

𝔽_{2}[d_{8}, d_{12}](yeL, a_{7}, a_{11})

where d_{8} = v^{2} + vw + w^{2}, d_{12} = vw(v + w), a_{7} = vy^{2}(e + f) + wy^{2}f, and a_{11} = v^{2}y^{2}(e + f) + w^{2}y^{2}f.

Proof: The classes d_{8} and d_{12} in 5.2 are the restrictions to the subgroup 2_{I}^{4} of the classes d_{8} and d_{12} given above.

We have the following relations among the generators x, y, e, f:

\begin{aligned}xe & = yf \\ xf & = y(e + f) \end{aligned}  \begin{aligned}ye^{2} & = y^{2}e \\ yf^{2} & = y^{2}f. \end{aligned}

Thus we find that T acts on ⟨ye, yf⟩ non-trivially while it acts trivially on ⟨v, w⟩ and ⟨y^{2}e, y^{2}f⟩. Moreover, since yeL is the only element in the radical of dimension 5, it follows that yeL is invariant under the entire group 3^{2}: 2.

Hence,

(6.2)  {𝔽_{2}[v, w](ye, yf, y^{2}e, y^{2}f, yeL)}^{⟨T ⟩} = 𝔽_{2}[v, w](y^{2}e, y^{2}f, yeL).

The τ action has the form v ↦ w ↦ v + w, y^{2}e ↦ y^{2}f ↦ y^{2}(e + f). Consequently, the invariant subring

(6.3)  {𝔽_{2}[v, w](ye, yf, y^{2}e, y^{2}f, yeL)}^{3^{2}} = 𝔽_{2}[d_{8}, d_{12}](a_{7}, b_{7}, a_{11}, b_{11}, yeL, c_{12}yeL)

where a_{7} = vy^{2}(e + f) + wy^{2}f, b_{7} = vy^{2}f + wy^{2}e, a_{11} = v^{2}y^{2}(e + f) + w^{2}y^{2}f, b_{11} = v^{2}y^{2}f + w^{2}y^{2}e and c_{12} = v^{3} + w^{3} + v^{2}w.

Finally, we take into account the action of A. A fixes v, y^{2}f, a_{7}, and a_{11}, and takes w to w + v, y^{2}e to y^{2}e + y^{2}f, b_{7} to a_{7} + b_{7}, b_{11} to a_{11} + b_{11}, and c_{12} to c_{12} + d_{12}. The resulting ring of invariants is

(6.4)  𝔽_{2}[d_{8}, d_{12}](yeL, vy^{2}(e + f) + wy^{2}f, v^{2}y^{2}(e + f) + w^{2}y^{2}f)

as claimed.

The comments at the end of section 5 on Sq^{1} and the Bockstein spectral sequence imply that the radical in H^{∗}(J_{2}; 𝔽_{2}) is the entire 𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11}). Putting all this together we have the result

Theorem 6.5: There is an exact sequence

𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11})⟶H^{∗}(J_{2}; 𝔽_{2})⟶

𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}](1, c_{3}, R_{6}, X_{8}, Y_{8}, X_{9}, Y_{9}, c_{3}R_{6}, X_{12}, Y_{12}, X_{13}, Y_{13}, l_{15}, V_{18}, c_{3}V_{18})

⊕ 𝔽_{2}[d_{2}, d_{3}, d_{8}, d_{12}]h_{14} ⊕ 𝔽_{2}[d_{3}, d_{8}, d_{12}]h_{15}.

Here, Sq^{1}(c_{3}) = d_{2}^{2}, Sq^{1}(d_{2}) = d_{3}, Sq^{1}(X_{2i}) = X_{2i+1}, Sq^{1}(Y_{2i}) = Y_{2i+1}, and there are Bocksteins β_{4}(k_{5}) = R_{6}, β_{8}(a_{7}) = d_{8}, β_{8}(a_{11}) = d_{12}.

§7. The invariants H^{∗}(2^{4})^{3×A_{5}}

The invariants H^{∗}(2^{4})^{A_{5}} were computed by Adem and Milgram in [AM1 – Thm. 3.2]. They were shown to be freely and finitely generated by 60 classes over a polynomial subalgebra on four variables. The calculation was simplified by extending to 𝔽_{4}-coefficients by adjoining a cube root of unity ξ and changing to a new basis. In this way of writing the invariants the 60 free generators are certain polynomials in classes L, M, C, D, and E with degrees 3, 3, 8, 9, and 9 respectively, while the generators of the polynomial subalgebra have degrees 5, 5, 12 and 12, and are denoted by X, Y, Z, and W, respectively.

Then τ (see (5.1)) acts on H^{∗}(2^{4}; 𝔽_{4})^{A_{5}}, with all the classes above being eigenvectors. The classes C, Z, and W are fixed, the classes M, E, Y are eigenvectors for ξ, while the classes L, D, and X have eigenvalue ξ^{2}.

By writing

𝔽_{4}[Z, W, X, Y] = 𝔽_{4}[Z, W, XY, X^{3} + Y^{3}](1, X, Y, X^{2}, Y^{2}, Y^{3})

we can show that H^{∗}(2^{4}; 𝔽_{4})^{3×A_{5}} is freely and finitely generated over the polynomial subalgebra 𝔽_{4}[Z, W, XY, X^{3} + Y^{3}] by 120 classes. First, the 60 classes listed in [AM1-Table 2] are sorted according to the action of τ: 22 classes are fixed, 19 have eigenvalue ξ and 19 have eigenvalue ξ^{2}. Then the 22 fixed classes are multiplied by 1 or Y^{3}. The 19 classes of eigenvalue ξ are multiplied by X or Y^{2}, and the 19 classes of eigenvalue ξ^{2} are multiplied by X^{2} or Y. As examples the classes of degree less than or equal to 13 resulting from this process are listed below.

Dimension Module generators
0  1
6  LM
8  C, XM, Y L
9  L^{3}, M^{3}
11  XL^{2}, Y M^{2}
12  LE, MD, L^{2}M^{2}
13  X^{2}L, Y^{2}M

Theorem 7.1: The Poincaré series for H^{∗}(2^{4})^{3×A_{5}} is

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

where q(x) is the polynomial of degree 45

p(x) = 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}

Note that we can obtain generators for H^{∗}(2^{4}; 𝔽_{2})^{3×A_{5}} by using the explicit descriptions in [AM1] for the classes L, M, etc. in terms of ξ, and using that if A = A_{1} + ξA_{2} is invariant, then so are A_{1} and A_{2}. We will use the notation L = L_{1} +ξL_{2}, M = M_{1} +ξM_{2}, etc., but it turns out that L_{2} = M_{2} = L_{1} + M_{1}. So we have

\begin{aligned}L & = L_{1} + ξ(L_{1} + M_{1}), \\ M & = M_{1} + ξ(L_{1} + M_{1}), \\ D & = D_{1} + ξ(D_{1} + E_{1}), \\ E & = E_{1} + ξ(D_{1} + E_{1}), \\ X & = X_{1} + ξ(X_{1} + Y_{1}), \\ Y & = Y_{1} + ξ(X_{1} + Y_{1}), \\ Z & = Z_{1} + ξ(Z_{1} + W_{1}), \\ W & = W_{1} + ξ(Z_{1} + W_{1}), \\ C & = C_{1}, \end{aligned}

(that is, C_{2} = 0). Then, taking the product XY we find that X_{1}^{2} + X_{1}Y_{1} + Y_{1}^{2} is an invariant. The 120 module generators can similarly be rewritten, and the following are a list of the module generators of degree less than or equal to 13. Note that altogether there are 120 generators for the invariants H^{∗}(2^{4}; 𝔽_{2})^{3×A_{5}} as a module over

𝔽_{2}[Z_{1}, W_{1}, X_{1}^{2} + X_{1}Y_{1} + Y_{1}^{2}, X_{1}^{2}Y_{1} + X_{1}Y_{1}^{2}].
Dimension  Module Generators
0  1
6  L_{1}^{2} + L_{1}M_{1} + M_{1}^{2}
8  C_{1}, X_{1}L_{1} + Y_{1}M_{1} + Y_{1}L_{1}, X_{1}L_{1} + Y_{1}M_{1} + X_{1}M_{1}
9  L_{1}^{3} + L_{1}^{2}M_{1} + M_{1}^{3}, L_{1}^{3} + L_{1}M_{1}^{2} + M_{1}^{3}
11  X_{1}L_{1}^{2} + Y_{1}M_{1}^{2} + Y_{1}L_{1}^{2}, X_{1}L_{1}^{2} + Y_{1}M_{1}^{2} + X_{1}M_{1}^{2}
12  L_{1}D_{1} + M_{1}E_{1} + M_{1}D_{1}, L_{1}D_{1} + M_{1}E_{1} + L_{1}E_{1}, L_{1}^{4} + L_{1}^{2}M_{1}^{2} + M_{1}^{4}
13  X_{1}^{2}L_{1} + Y_{1}^{2}M_{1} + X_{1}^{2}M_{1}, X_{1}^{2}L_{1} + Y_{1}^{2}M_{1} + Y_{1}^{2}L_{1}

Theorem 7.2: The intersection of the image of restriction from the Sylow 2-subgroup S_{J} to 2^{4} with the invariants

Im(res_{2^{4}}^{S_{J}}) ∩ H^{∗}(2^{4}; 𝔽_{2})^{3×A_{5}}

is freely and finitely generated over the polynomial subalgebra

𝔽_{2}[Z_{1} + W_{1}, Z_{1}W_{1}, X_{1}^{2} + X_{1}Y_{1} + Y_{1}^{2}, X_{1}^{2}Y_{1} + X_{1}Y_{1}^{2}]

by 237 classes.

Proof: The classes L_{1} and M_{1} are the same as the classes L and M given in Proposition 5.6. Then X_{1} = Sq^{2}(L_{1}), Y_{1} = Sq^{2}(M_{1}), D_{1} = Sq^{4}(X_{1}), and E_{1} = Sq^{4}(Y_{1}) are also in the image of restriction. The class C_{1} was shown to be in the image of restriction using Mathematica:

C_{1} = Res_{2^{4}}^{S_{J}} (w_{8} + LN_{2} + LN_{1} + MN_{1} + v_{4}(yL + xM + yM) + v_{4}^{2} + x^{8} + x^{4}y^{4} + y^{8}).

This implies that all 120 module generators mentioned in the description of H^{∗}(2^{4})^{3×A_{5}} are in the image of restriction, as well as the polynomial generators X_{1}^{2} + X_{1}Y_{1} + Y_{1}^{2} and X_{1}^{2}Y_{1} + X_{1}Y_{1}^{2}. The fact that Z_{1} + W_{1} and Z_{1}W_{1} are in the image of restriction, but not Z_{1} or W_{1}, followed from another computer calculation. For example

Z_{1} + W_{1} = Res_{2^{4}}^{S_{J}} (yLw_{8} + xLw_{8} + xMw_{8} + v_{4}w_{8}+
x^{2}y^{2}LN_{1} + y^{4}LN_{1} + v_{4}LN_{1} + v_{4}LN_{2}
+ v_{4}MN_{1} + ¸(x^{4} + x^{3}y + y^{4})(LN_{1} + LN_{2} + MN_{1})
+ (x^{5}y^{2} + x^{4}y^{3} + x^{2}y^{5} + xy^{6})N_{1}
+ (x^{4}y^{2} + x^{2}y^{4})(L^{2} + LM + yN_{1}) + v_{4}^{2}(xL + yM)
+ v_{4}(x^{5} + xy^{4} + y^{5})L + v_{4}y^{5}M + (x^{6}y^{3} + x^{2}y^{7})L
+ v_{4}^{2}(xL + yL + xM) + x^{4}v_{4}(xL + yL + xM)
+ (x^{5}y^{3} + x^{4}y^{4} + x^{3}y^{5} + x^{2}y^{6})(xL + yL + xM)
+ (x^{4}y^{4} + x^{2}y^{6})v_{4} + x^{8}y^{4} + x^{4}y^{8}).

Since we can write 𝔽_{2}[Z_{1}, W_{1}] as 𝔽_{2}[Z_{1} + W_{1}, Z_{1}W_{1}](1, Z_{1}), we must consider the product of Z_{1} with the 120 invariant classes. A computer calculation showed that the classes Z_{1}X_{1}, Z_{1}Y_{1}, Z_{1}D_{1}, Z_{1}E_{1}, Z_{1}L_{1}^{2}, Z_{1}L_{1}M_{1}, and Z_{1}M_{1}^{2} are in the image of restriction, although the classes Z_{1}L_{1} and Z_{1}M_{1} are not. This implies that 117 of the 120 products with Z_{1} are in the image of restriction, the only exceptions being the product with 1, C_{1}, and C_{1}^{2}. The fact that Z_{1}, Z_{1}C_{1}, and Z_{1}C_{1}^{2} are not in the image follows from restricting to the 2^{2} subgroup which is the center of the 2^{2+4}, and noting that these restrictions are not invariant under an S_{3} action. Only the classes Z_{1}, W_{1}, and C_{1} restrict nontrivially to this 2^{2} subgroup. This last observation is what implies the equality mentioned in the introduction between the image of restriction res_{2^{4}}(H^{∗}(J_{3})) with the intersection of the invariants (H^{∗}(2^{4}))^{3×A_{5}} and the set L of classes in H^{∗}(2^{4}) which restrict to (H^{∗}(2^{2}))^{2}.

Theorem 7.3: The radical of H^{∗}(J_{3}; 𝔽_{2}) is isomorphic to the radical of H^{∗}(2^{2+4}; 𝔽_{2})^{3^{2}:2},

𝔽_{2}[d_{8}, d_{12}](k_{5}, a_{7}, a_{11}).

Proof: We consider the Bockstein spectral sequence for H^{∗}(J_{3}). Computing Sq^{1} for classes in H^{∗}(2^{4}; 𝔽_{2})^{3×A_{5}} yields Sq^{1}(L_{1}) = Sq^{1}(M_{1}) = 0, Sq^{1}(X_{1}) = L_{1}^{2}, Sq^{1}(Y_{1}) = M_{1}^{2}, Sq^{1}(D_{1}) = X_{1}^{2}, and Sq^{1}(E_{1}) = Y_{1}^{2}. Thus for our module generators of degree less than or equal to 13, we have

Sq^{1}(L_{1}^{2} + L_{1}M_{1} + M_{1}^{2}) = 0,
Sq^{1}(X_{1}L_{1} + Y_{1}M_{1} + Y_{1}L_{1}) = L_{1}^{3} + L_{1}M_{1}^{2} + M_{1}^{3},
Sq^{1}(X_{1}L_{1} + Y_{1}M_{1} + X_{1}M_{1}) = L_{1}^{3} + L_{1}^{2}M_{1} + M_{1}^{3},
Sq^{1}(X_{1}L_{1}^{2} + Y_{1}M_{1}^{2} + Y_{1}L_{1}^{2}) = Sq^{1}(X_{1}L_{1}^{2} + Y_{1}M_{1}^{2} + X_{1}M_{1}^{2})
= L_{1}^{4} + L_{1}^{2}M_{1}^{2} + M_{1}^{4},
Sq^{1}(L_{1}D_{1} + M_{1}E_{1} + M_{1}D_{1}) = X_{1}^{2}L_{1} + Y_{1}^{2}M_{1} + X_{1}^{2}M_{1},
Sq^{1}(L_{1}D_{1} + M_{1}E_{1} + L_{1}E_{1}) = X_{1}^{2}L_{1} + Y_{1}^{2}M_{1} + Y_{1}^{2}L_{1}.

A computer calculation also yields Sq^{1}(C_{1}) = Sq^{1}(X_{1}M_{1} + Y_{1}L_{1}). We denote D_{8} = C_{1} + X_{1}M_{1} + Y_{1}L_{1}.

Turning to the polynomial generators, we have Sq^{1}(X_{1}^{2} +X_{1}Y_{1} +Y_{1}^{2}) = X_{1}M_{1}^{2} +Y_{1}L_{1}^{2} and Sq^{1}(Z_{1} + W_{1}) = Sq^{1}(L_{1}E_{1} + M_{1}D_{1}). We write D_{12} = Z_{1} + W_{1} + L_{1}E_{1} + M_{1}D_{1}.

Thus through degree 13 we have surviving classes in degree 6, 8 and 12 only. This implies that the classes k_{5}, a_{7}, and a_{11} must be in the image of restriction from J_{3} to 2^{2+4}.

The classes D_{8} and D_{12} defined above are not equal to the classes d_{8} and d_{12} in 5.2, but the restrictions to V = 2^{2} = ⟨z, t⟩ are equal. Thus H^{∗}(J_{3}; 𝔽_{2}) contains the polynomial algebra on the generators D_{8} and D_{12}. The relations in H^{∗}(2^{2+4}; 𝔽_{2}) now imply that the products with the nilpotent classes k_{5}, a_{7}, and a_{11} are equal, for example d_{8}k_{5} = D_{8}k_{5}. Thus the radical of H^{∗}(J_{3}; 𝔽_{2}) is as stated.

This completes the description of H^{∗}(J_{3}; 𝔽_{2}).

§8. The Quillen complex for J_{2}

The 2-local subgroups of J_{2} are classified in Janko [J] and Finkelstein and Rudvalis [FR]. There are three maximal 2-local subgroups: 2_{−}^{1+4}: A_{5} (the centralizer of a central involution), 2^{2+4}: (3×S_{3}) (the normalizer of a Klein four group containing three involutions of central type), and A_{4} × A_{5} (the normalizer of one of the conjugacy classes of a Klein four group containing three involutions of non-central type).

We determine the quotient of the Quillen complex Δ by J_{2}. The papers of Gorenstein and Harada [GH] and Finkelstein and Rudvalis [FR] were useful in determining the following facts.

Let S_{J} denote a Sylow 2-subgroup of J_{2}. Then S_{J} contains 35 involutions, which in J_{2} fall into two conjugacy classes, 11 type A or central and 24 type B or non-central. S_{J} contains 73 Klein four groups, which in J_{2} fall into four conjugacy classes, 73 = 5 + 36 + 8 + 24. We denote these 2^{2}-groups by A^{3}, A^{1}B^{2}, B_{I}^{3} and B_{II}^{3}. This notation describes the involutions as central or non-central, but there are two conjugacy classes of type B^{3}.

S_{J} contains 30 eights groups 2^{3}, which in J_{2} fall into two conjugacy classes 30 = 6+24, denoted by A^{3}B^{4} and A^{1}B^{6}. S_{J} contains 2 sixteens groups 2^{4}, denoted A^{3}B^{12}, which are conjugate in S_{J}. Thus the quotient of the Quillen complex Δ/J_{2} contains nine vertices, one for each conjugacy class of elementary abelian 2-subgroup in J_{2}. The normalizers of each of these nine elementary abelian 2-subgroups are listed below.

(8.1)  \begin{aligned}N_{J_{2}}(A^{1}) & = 2_{−}^{1+4}: A_{5} \\ N_{J_{2}}(B^{1}) & = 2 × 2 × A_{5} ⊂ N_{J_{2}}(B_{I}^{3}) \\ N_{J_{2}}(A^{3}) & = 2^{2+4}: (3 × S_{3}) \\ N_{J_{2}}(A^{1}B^{2}) & = 2^{4}: 2 ⊂ S_{J} \\ N_{J_{2}}(B_{I}^{3}) & = A_{4} × A_{5} \\ N_{J_{2}}(B_{II}^{3}) & = 2^{4}: 3 ⊂ N_{J_{2}}(A^{3}) \\ N_{J_{2}}(A^{3}B^{4}) & = 2^{2+4}: 3 ⊂ N_{J_{2}}(A^{3}) \\ N_{J_{2}}(A^{1}B^{6}) & = A_{4} × 2 × 2 ⊂ S_{J}: 3 \\ N_{J_{2}}(A^{3}B^{12}) & = 2^{2+4}: (3 × 3) ⊂ N_{J_{2}}(A^{3}). \end{aligned}

See sections 1 and 2 for notations for the generators of S_{J} and for the 3×3 in N_{J_{2}}(A^{3}). We choose representatives for each conjugacy class of elementary abelian subgroup as follows: A^{1} = ⟨z⟩, B^{1} = ⟨x⟩, A^{3} = ⟨z, t⟩, A^{1}B^{2} = ⟨z, x⟩, B_{I}^{3} = ⟨x, y⟩, B_{II}^{3} = ⟨x, yz⟩, A^{3}B^{4} = ⟨z, t, x⟩, A^{1}B^{6} = ⟨z, x, y⟩, and A^{3}B^{12} = ⟨z, t, x, y⟩.

Note that N_{J_{2}}(B_{II}^{3}) contains the order three element fT τ^{−1}, which permutes the three elements of {x, yz, xyz}.

The references [Atlas, FR, J] give the normalizers of A^{1}, B^{1}, A^{3}, and B_{I}^{3}, and [FR] shows that N_{J_{2}}(B_{II}^{3}) ⊂ N_{J_{2}}(A^{3}). It is also obvious that the normalizers of A^{1}B^{2} and A^{1}B^{6} are contained in N_{J_{2}}(A^{1}), and that the normalizers of A^{3}B^{4} and A^{3}B^{12} are contained in N_{J_{2}}(A^{3}), by considering the action on the A-pure part.

Note that N_{J_{2}}(A^{1}B^{2}) contains the element e. Given any n in this normalizer, we may assume that n fixes x, by multiplying by e if necessary. Then n is an element of N_{J_{2}}(A^{1}) ∩ N_{J_{2}}(B^{1}) = 2^{4}. Thus N_{J_{2}}(A^{1}B^{2}) = 2^{4}: 2.

Similarly, since fT τ^{−1} ∈ N_{J_{2}}(B_{II}^{3}), we may assume that any element n fixes x, and so n ∈ 2 × 2 × A_{4}. Since τ does not normalize B_{II}^{3}, we actually have n ∈ 2^{4}. Thus N_{J_{2}}(B_{II}^{3}) = 2^{4}: 3.

Next, since T ∈ N_{J_{2}}(A^{1}B^{6}), we may assume that any element n, after multiplying by T if necessary, also normalizes the subgroup A^{1}B^{2}, so n ∈ 2^{4}: 2. But e does not normalize A^{1}B^{6}, and so n ∈ 2^{4}. Thus N_{J_{2}}(A^{1}B^{6}) = 2^{4}: 3 = A_{4} × 2 × 2.

The normalizer N_{J_{2}}(A^{3}B^{4}) contains 2^{2+4}: 3, with 3 = ⟨τ⟩, and is contained in N_{J_{2}}(A^{3}). Note that the remaining S_{3} ⊂ N_{J_{2}}(A^{3}) is generated by ⟨T τ^{−1}, A⟩. None of these elements normalize A^{3}B^{4}.

Finally, it is clear that N_{J_{2}}(A^{3}B^{12}) contains 2^{2+4}: (3 × 3), but is not all of N_{J_{2}}(A^{3}) since the 2^{4} is not normal in the Sylow 2-subgroup S_{J}.

We have thus described all of the normalizers of the representative elementary abelian 2-groups. Next, the flags, or chains of inclusions, of the elementary abelian 2-subgroups of J_{2} must be studied, and the flag stabilizers must be determined. The quotient of the Quillen complex contains 23 edges, 25 faces, and 10 quads. The conjugacy classes of flags are determined by their vertex types except for the following.

There are two conjugacy classes of flags of type B^{1} ⊂ A^{1}B^{6}. We choose representatives for these flags as ⟨x⟩ ⊂ ⟨z, x, y⟩ and ⟨xz⟩ ⊂ ⟨z, x, y⟩. Thus in the quotient of the Quillen complex Δ/J_{2}, there are two edges between the vertices B^{1} and A^{1}B^{6}.

There are two conjugacy classes of flags of type B^{1} ⊂ A^{1}B^{2} ⊂ A^{1}B^{6}, the “join” of the two edges B^{1} ⊂ A^{1}B^{6} with the vertex A^{1}B^{2}. There are two conjugacy classes of flags of type B^{1} ⊂ A^{1}B^{6} ⊂ A^{3}B^{12}, the join of the two edges with the vertex A^{3}B^{12}. More generally, the complex Δ/J_{2} is the join of the vertex A^{3}B^{12} with a subcomplex (every flag is conjugate to one contained in A^{3}B^{12}).

There are three conjugacy classes of flags of type B^{1} ⊂ B_{II}^{3} ⊂ A^{1}B^{6}, described by A^{1}B^{6} = ⟨z, x, y⟩, B_{II}^{3} = ⟨x, yz⟩, and the three possible involutions B^{1} ∈ {x, yz, xyz}.

There are two conjugacy classes of flags of type B^{1} ⊂ A^{1}B^{2} ⊂ A^{1}B^{6} ⊂ A^{3}B^{12}, and three conjugacy classes of flags of type B^{1} ⊂ B_{II}^{3} ⊂ A^{1}B^{6} ⊂ A^{3}B^{12}.

The stabilizers of all 67 cells in Δ/J_{2} are easily computed. Webb’s theorem [W1,W2] always applies to the Quillen complex of any group, so we obtain an Euler characteristic formula containing 67 terms. There are many repetitions among the terms and cancellations due to opposite signs. We obtain the following formula, which is valid for the 2-primary component of cohomology with coefficients in any ℤ[J_{2}]-module M.

(8.2) \begin{aligned}H^{n}(J_{2}; M)_{2} = & H^{n}(2_{−}^{1+4}: A_{5}; M)_{2} + H^{n}(2^{2+4}: (3 × S_{3}); M)_{2} \\ & − H^{n}(S_{J}: 3; M)_{2} + H^{n}(A_{4} × A_{5}; M)_{2} − H^{n}(A_{4} × A_{4}; M)_{2}. \end{aligned}

§9. Some remarks on the computer calculations

Part of the calculation of the cohomology of the Janko groups was achieved using computer generations of projective resolutions, cup products and restrictions. Many of the programs for the calculation have been used in other settings and are adequately described in the papers [CGS] and [Car]. The new wrinkle for these calculations is a complete implementation of restriction maps for the mod(p) cohomology rings of p-groups. In this section we present a brief description of the process. All of the programs are written in the MAGMA language and are suppported by the MAGMA computer algebra system. However stand-alone versions are under construction [DG].

Throughout the section let G denote a finite p-group and let k = 𝔽_{p} be the prime field of characteristic p. All cohomology calculation for kG are preceded by the construction of a standard (i. e. fixed) free module. With this construction, the elements of the group algebra can be identified in a retrievable way with vectors in k^{n} for n = |G|, the order of G. So if P = ∑_{1}^{a} kGα_{i} and Q = ∑_{1}^{b} kGβ_{j} are free modules with kG-bases {α_{i}} and {β_{j}} respectively, then a homomophism

θ : P ⟶Q

is determined entirely by the vectors v_{i,j} ∈ k^{n} where θ(α_{i}) = ∑_{1}^{b} v_{i,j}β_{j}. Hence a projective resolution (P_{∗}, ϵ) of a kG-module can be stored as rows of a matrix with |G| columns.

The actual restriction map is realized through a chain map on projective resolutions. That is, suppose that (P_{∗}, ϵ) is a minimal kG-projective resolution of the trivial kG-module k and that (Q_{∗}, τ) is a minimal kH-projective resolution of k. What is needed is a degree-0 chain map

μ : Q_{∗}⟶P_{∗}

which lifts the identity map on k. The first step is to convert the resolution (P_{∗}, ϵ) into a resolution of kH-modules. In particular, each row of the matrix which records the boundary homomorphisms in the resolution is a vector representing an element of kG and this must be converted to a sequence of vectors representing the corresponding elements of kH. Some further discussion of this is included later.

For the moment we assume that (P_{∗}, ϵ) is the restriction of the kG-resolution to a kH-resolution. Then finding the chain map at each stage is a straightforward process involving solving sets of linear equations. That is, if we have the chain map to degree n, then we are looking at the following diagram

⋯ ⟶ Q_{n+1} ⟶^{∂′_{n+1}} Q_{n} ⟶ ⋯
↓μ_{n}
⋯ ⟶ P_{n+1} ⟶^{∂_{n+1}} P_{n} ⟶ ⋯

Consequently, construction at degree n + 1 is a matter of solving the system of linear equations (m)∂′_{n+1}μ_{n} = (x)∂_{n+1} for all of the generators m of Q_{n+1}. Notice here that we are writing our maps on the right because the computer prefers to do row operations.

Once the chain map μ has been created, it is an easy task to get the restriction of the generators of the kG-cohomology. An element ζ ∈ H^{n}(G, k) is represented by a cocyle ζ′ ∈ Hom_{kG}(P_{n}, k) and it is straightforward to express the composition

ζ′μ ∈ Hom_{kH}(Q_{n}, k) ≅ H^{n}(H, k)

as a polynomial in the generators of the cohomology ring of H.

One of the most time consuming parts of the program is the conversion of the minimal kG-resolution (P_{∗}, ϵ) to a (no longer minimal) kH-resolution. As we stated, the boundary homomorphisms in (P_{∗}, ϵ) are given as sequences of blocks

θ_{i,j} : kG⟶kG

where (1)θ_{i,j} = v_{i,j} as before. Then given a kH-basis,

{γ_{i}|i = 1 … |G : H|}

of kG, it is a matter of finding the elements {w_{a,b}} of kH such that (γ_{a})θ_{i,j} = ∑ w_{a,b}γ_{b}. The current implementation of the technique actually creates the matrix of the map θ_{i,j} from the element v_{i,j} ∈ kG, then performs a similarity transformation on the matrix to get it into a form from which the elements {w_{a,b}} can be read off as parts of row vectors. Because the basis {γ_{i}|i = 1 … |G : H|} is fixed for all of the restrictions to H the matrix of the similarity transformation is also fixed and only needs to be calculated once.

The main use of the computer was in calculating the restrictions of the generators out to degree 6 of the groups S_{J} and 2^{2+4} to the three nonconjugate subgroups of type Q_{8} × 2 and to the two elementary abelian 2^{4}’s in 2^{2+4}. The basic results are summarized in (4.4).

The calculations were performed on a SPARC 2 and required less than 10M. of memory. The projective resolution for S_{J} required approximately 4.5 minutes of computer time, while the chain map program to get the generators of H^{∗}(S_{J}, 𝔽_{2}) in degrees up to degree 6, took under 9 minutes. The corresponding times for the cohomology of the 2^{2+4} were approximately 2 minutes and 4 minutes. The complete calculations of the projective resolutions to degree 6 and the generators and relations for the cohomology rings of the subgroups Q_{8} × 2 and 2^{4} were accomplished in only seconds of computing time. Each of the restrictions requires between 5 and 8 minutes. Approximately 2 minutes of the time for each restriction was expended in converting the resolution of the trivial module for the larger group to a resolution for the subgroup.

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Jon Carlson Department of Mathematics University of Georgia Athens, GA, 30602

e-mail: jfc@sloth.math.uga.edu

John Maginnis

Department of Mathematics Kansas State University Manhattan, KA, 66506

e-mail: maginnis@math.ksu.edu

R. James Milgram

Department of Mathematics Stanford University Stanford, CA, 94305

e-mail: milgram@math.stanford.edu