The Cohomology of the Lyons Group and Double Covers of Alternating Groups

Alejandro Adem† Dikran Karagueuzian‡ Department of Mathematics University of Wisconsin, Madison WI 53706

R. James Milgram† Department of Mathematics Stanford University, Stanford CA 94305

Kristin Umland Department of Mathematics University of New Mexico, Albuquerque NM 87131

§0 INTRODUCTION

From the Classification Theorem for Finite Simple Groups we know that there are precisely eight sporadic simple groups which have (ℤ/2)^{4} as their largest elementary abelian 2–subgroup (i.e. they have rank equal to four at the prime p = 2): the two Mathieu groups M_{22}, M_{23}; the two Janko groups J_{2}, J_{3}; the McLaughlin group McL; the Lyons group Ly; and two groups with much larger 2-Sylow subgroups HS (Higman–Sims) and Co_{3} (one of Conway’s groups). In previous work the mod 2 cohomology rings of M_{22} [AM2], M_{23} [M], J_{2} and J_{3} [CMM], and McL [AM3] were computed. One of the main objectives of this paper is to determine the cohomology ring H^{∗}(Ly; 𝔽_{2}).

Even though Ly is a large and complicated group (it is the largest of those in the list of above, having order equal to 51, 765, 179, 004 × 10^{6}), the answer turns out to be surprisingly simple. We show that Ly has two conjugacy classes of maximal 2-elementary subgroups, one 2^{4} with Weyl group A_{7}, and one 2^{3} with Weyl group L_{3}(2), and we prove that these subgroups detect H^{∗}(Ly). In [AM4] we determined the ring of invariants

H^{∗}(2^{4})^{A_{7}} = 𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}](1, a_{18}, a_{20}, a_{21}, a_{24}, a_{25}, a_{27}, a_{45})

while it is well–known that the invariants H^{∗}(2^{3})^{L_{3}(2)} are 𝔽_{2}[d_{4}, d_{6}, d_{7}], the Dickson algebra. From this we obtain Theorem 7.3

† Partially supported by a grant from the NSF.

‡ Supported by an NSF Postdoctoral Fellowship.

There is a short exact sequence of rings

0⟶H^{∗}(Ly; 𝔽_{2})⟶H^{∗}(2^{4})^{A_{7}} ⊕ 𝔽_{2}[d_{4}^{2}, d_{6}^{2}, d_{7}](1, d_{4}d_{7}, d_{6}d_{7}, d_{4}d_{6}d_{7})⟶𝔽_{2}[h_{8}, h_{12}]⟶0

with h_{8} the image of d_{8} in the A_{7} invariants and d_{4}^{2} in the L_{3}(2) invariants, while h_{12} is the image of d_{12} and d_{6}^{2} respectively.

Thus we can write

H^{∗}(Ly, 𝔽_{2}) ≅ H^{∗}(2^{4})^{A_{7}} ⊕ d_{7}𝔽_{2}[d_{4}, d_{6}, d_{7}]

where the role of d_{7} and d_{4}d_{7} is to have ℤ/8–Bocksteins to d_{8} and d_{12} respectively. In the course of obtaining this result we determine the cohomology of a number of groups which arise in other contexts and perhaps our results for these further groups are even more intriguing.

The double covers of the alternating groups 2A_{n} arise in studying the two connected cover of the infinite loop space Q(S^{0}), which can actually be identified with the plus construction B_{2A_{∞}}^{+}. The mod 2 cohomology of B_{2A_{∞}} is easily determined since H^{∗}(B_{A_{∞}}, 𝔽_{2}) is known as a polynomial algebra on a specified but infinite set of generators, and the elements σ_{2}, Sq^{1}(σ_{2}), Sq^{2}Sq^{1}(σ_{2}), …, Sq^{2^{n}}Sq^{2^{n−1}} ⋯ Sq^{2}Sq^{1}(σ_{2}), …, form a regular sequence where σ_{2} is the non-zero element in dimension two. However, the cohomology of the individual 2A_{n}’s is much more complex and the inclusions 2A_{n} ↪ 2A_{n+1} tend to often have very large cokernels in cohomology.

By studying these cokernels it appears that they give information related to the structure of periodicity operators in homotopy theory, and (as one sees in the proof of 4.2 where the exact structure of the cokernel for 2A_{8} is seen to be equivalent to the determination of the ring of invariants) also appear to play an intriguing role in modular invariant theory.

Moreover, 2A_{11} is a maximal, odd index subgroup of Ly while the Sylow 2-subgroup of 2A_{8} is the Sylow 2-subgroup of M_{22}, M_{23} and McL. Our route to the cohomology of Ly determines the cohomology of the following groups in succession

2A_{8} ↪ 2S_{8} ↪ 2A_{10} ↪ 2A_{11}

where the last inclusion is a 2-local equivalence. We now describe how this goes.

It is well–known that H^{2}(A_{n}, 𝔽_{2}) ≅ 𝔽_{2}, hence for each of them there exists a unique non–trivial double cover

1 → ℤ/2 → 2A_{n} → A_{n} → 1.

An interesting group–theoretic fact is that for n ≤ 7, 2A_{n} does not contain ℤ/2 × ℤ/2 as a subgroup. Hence H^{∗}(2A_{n}, 𝔽_{2}) is periodic in this range, and in fact they are all isomorphic to 𝔽_{2}[u_{4}] ⊗ Λ(x_{3}), a polynomial algebra on a 4–dimensional generator tensored with an exterior algebra on a 3–dimensional class. This pleasant behaviour breaks down when n = 8 and in fact the calculation of H^{∗}(2A_{8}, 𝔽_{2}) has been an open problem for some time. Attempts to use the description of H^{∗}(A_{8}, 𝔽_{2}) provided in [AMM] have been unsuccessful due to the a priori complicated nature of the spectral sequence associated to the extension above. We show:

Theorem 3.3 The mod 2 cohomology of 2A_{8} can be described by a long exact sequence

0 → 𝔽_{2}[d_{4}, d_{8}](u_{3}, u_{7}, u_{9}) → H^{∗}(2A_{8}, 𝔽_{2}) → H^{∗}(E)^{L_{3}(2)} ⊕ H^{∗}(F)^{L_{3}(2)} → 𝔽_{2}[d_{4}, d_{8}] → 0

where E and F are rank 4 elementary abelian subgroups in 2A_{8}, and

H^{∗}(E)^{L_{3}(2)} = H^{∗}(F)^{L_{3}(2)} = 𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}](1, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{21}).

In the expression above, E and F represent the two conjugacy classes of maximal elementary abelian subgroups. The term on the left is the radical (i.e. the nilpotent elements) in H^{∗}(2A_{8}). The term on the right represents the double image classes. We use the above to compute H^{∗}(2S_{8}, 𝔽_{2}), where 2S_{8} is the double cover of S_{8} which extends the double cover of the alternating group. Unexpectedly, it turns out that the elementary abelian subgroups detect the cohomology of this double cover. More precisely we have

Theorem 5.4 The cohomology of 2S_{8} is determined by the following exact detection sequence

\begin{matrix}0 → H^{∗}(2S_{8}, & 𝔽_{2}) → 𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}](1, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{21}) \\ ⊕ 𝔽_{2}[w_{1}, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}) → 𝔽_{2}[h_{4}, h_{8}] → 0 \end{matrix}

where h_{4}, h_{8} correspond to the double image classes (d_{4}, d_{2}^{2}) and (d_{8}, d_{4}^{2}) respectively.

One can deduce from this that the spectral sequence for the extension 2A_{8} ◁ 2S_{8} collapses at E_{2}; the notation reflects this fact. Here we have once again two conjugacy classes of maximal elementary abelian subgroups, one of rank 4 (obtained by fusing the 2 representatives for 2A_{8}) and one of rank 3. As before, the term on the right represents the double image classes.

Noting that 2S_{8} ⊂ 2A_{10} is an odd index subgroup, we infer that H^{∗}(2A_{10}, 𝔽_{2}) is also detected on elementary abelian subgroups. The maximal elementary abelian subgroups are weakly closed in 2S_{8} ⊂ 2A_{10} and hence, using the Cárdenas–Kuhn Theorem (see [AM1]) the image of the restriction maps can be obtained as the intersection of the restriction from H^{∗}(2S_{8}, 𝔽_{2}) with the invariants under the appropriate Weyl groups: still L_{3}(2) for the 2^{4} and now S_{4} (instead of D_{8}) for the 2^{3}. In fact it can be described as follows.

Theorem 6.1 The cohomology of 2A_{10} is determined by the following exact detection sequence

0 → H^{∗}(2A_{10}, 𝔽_{2}) → H^{∗}(2^{4})^{L_{3}(2)} ⊕ 𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}) → 𝔽_{2}[h_{4}, h_{8}] → 0

where h_{4}, h_{8} correspond to the double image classes (d_{4}, d_{2}^{2}) and (d_{8}, d_{4}^{2}) respectively.

To obtain H^{∗}(Ly, 𝔽_{2}) from the expression above, we simply need to take A_{7} invariants and compute the intersection of the image of H^{∗}(2A_{10}) in H^{∗}(2^{3}) with the Dickson Algebra of L_{3}(2)–invariants.

It would seem that Theorem 5.4 should admit an extension to double covers of S_{2^{i}} for i ≥ 4. These groups carry important homotopy–theoretic data in their cohomology and hence deserve further attention. We will defer this to a sequel.

This paper is organized as follows: in §1 we discuss preliminaries; in §2 we determine the double image classes for the cohomology of 2A_{8}; in §3 we determine the radical of H^{∗}(2A_{8}); in §4 we consider the spectral sequence associated to the central extension 1 → ℤ/2 → 2A_{8} → A_{8} → 1; in §5 we describe facts about the 2–Sylow subgroup of Ly and determine the cohomology of 2S_{8}; in §6 we determine the cohomology of 2A_{10}; and in §7 we calculate H^{∗}(Ly, 𝔽_{2}).

We are grateful to L. Evens, M. Isaacs and S. Priddy for helpful comments. We thank Gregor Kemper for the use of a beta test version of his invariant theory package, and for the method of argument with which we determined H^{∗}(2^{4})^{L_{3}(2)}. Some of the calculations in this paper were first performed or subsequently checked using Magma ([BCP]). We thank the Magma project for their assistance in the use of this valuable tool. We thank R. Lyons for providing a purely group-theoretic proof of Corollary 5.5. Also we should point out that the first indication that such detection results might be possible occurs in [U] where, besides determining the Weyl group of the new 2^{3} in Ly, it was shown that the element u_{7} when looked at in H^{∗}(Ly, 𝔽_{2}) was not nilpotent.

Throughout this paper, coefficients will be assumed taken in the field 𝔽_{2} with two elements, so they are suppressed.

§1. PRELIMINARIES

We begin by discussing the mod 2 cohomology of the double covers 2A_{n} for n ≤ 7. In this range all these groups have periodic cohomology rings at the prime 2, and they can be easily computed. In fact we have the isomorphisms 2A_{4} ≅ SL_{2}(𝔽_{3}), 2A_{5} ≅ SL_{2}(𝔽_{5}), 2A_{6} ≅ SL_{2}(𝔽_{9}), while a simple double coset argument shows that 2A_{7} is mod 2 cohomologous to 2A_{6}. Note that for the same reason 2A_{4} is mod 2 cohomologous to 2A_{5}. We state their cohomology, leaving details to the reader (or see [AM1], Ch.IV):

Proposition 1.0 If G = 2A_{n}, where n ≤ 7, then H^{∗}(G) ≅ 𝔽_{2}[v_{4}] ⊗ E(y_{3}).

This remarkably simple behaviour breaks down rather dramatically for n = 8. Even though the mod 2 cohomology of A_{8} was computed in [AMM] some time ago, the computation for its double cover poses serious technical difficulties. Our strategy will be to determine H^{∗}(2A_{8}) using techniques developed to study the cohomology of certain sporadic simple groups. This calculation in turn will allow us to compute the cohomology of 2A_{10} and from there that of the sporadic simple group Ly.

To begin we provide background information needed to compute the mod 2 cohomology of the group G = 2A_{8}. First we recall that G has precisely two conjugacy classes of maximal elementary abelian 2–subgroups, both of rank 4, which from now on we will denote by E and F (see [AM3]). These are constructed as follows. First consider the regular representation of (ℤ/2)^{3} = V_{3} ⊂ S_{8}. This representation has Weyl group Aut((ℤ/2)^{3}) = GL_{3}(2). (As is always the case for the regular representation G ↪ S_{|G|}, we have N(G)/G = Out(G).) However, restricting V_{3} to A_{8} gives two non-conjugate subgroups V_{3}, V′_{3} in A_{8} which fuse in S_{8}, and both V, V′ lift to (ℤ/2)^{4}’s in 2A_{8}. In fact, if H = Syl_{2}(2A_{8}), then it contains precisely two copies of (ℤ/2)^{4}. The subgroups E and F intersect in a rank 2 subgroup V = E ∩ F and Lemma 1.1:

The normalizers of E, F in 2A_{8} are split extensions of the form E : L_{3}(2) and F : L_{3}(2), which are isomorphic since the two (ℤ/2)^{4}’s fuse in 2S_{8} = (2A_{8}): 2. In particular both of these maximal 2-tori have L_{3}(2) as their Weyl group.

The action of these matrix groups on E, F can be written down explicitly as follows: L_{3}(2) is the subgroup of L_{4}(2) generated by the following two matrices

A = ( \begin{matrix}1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 \end{matrix} ) B = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{matrix} )

with simple components the usual action of L_{3}(2) on (ℤ/2)^{3} and the trivial action on ℤ/2. But the representation is non-split and this will be reflected in the structure of the ring of invariants H^{∗}((ℤ/2)^{4})^{L_{3}(2)}.

Recall that the mod(2) cohomology of 2^{4}, H^{∗}(2^{4}), is the polynomial ring on four one dimensional variables, 𝔽_{2}[x_{1}, y_{1}, z_{1}, w_{1}] and the action of L_{3}(2) is given by the transposes of A and B above. Consequently, the subring 𝔽_{2}[x_{1}, y_{1}, z_{1}] is invariant under the action and the Dickson algebra 𝔽_{2}[x_{1}, y_{1}, z_{1}]^{L_{3}(2)} = 𝔽_{2}[d_{4}, d_{6}, d_{7}] ⊂ 𝔽_{2}[x_{1}, y_{1}, z_{1}, w_{1}]^{L_{3}(2)}. In fact, by a standard computer assisted computation of invariants, we have

Proposition 1.2:

The ring of invariants 𝔽_{2}[x_{1}, y_{1}, z_{1}, w_{1}]^{L_{3}(2)} can be described as

𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}](1, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{21})

where the subscript denotes the dimensions of the generators. The ring is freely generated over the polynomial subring spanned by d_{4}, d_{6}, d_{7}, d_{8} and furthermore we have a_{9} = Sq^{1}a_{8}, a_{11} = Sq^{2}Sq^{1}a_{8}, a_{12} = Sq^{2}a_{10}, a_{13} = Sq^{1}Sq^{2}a_{10} and a_{21} = a_{10}a_{11}.

Proof: Write W ≅ L_{3}(2). It was shown in [AM2] that for a 2-Sylow subgroup P ⊂ W, the invariant ring H^{∗}(E)^{P} is Cohen-Macaulay; it therefore follows (see [B]) that H^{∗}(E)^{W} is Cohen-Macaulay also.

Using the modular invariant theory package developed by Gregor Kemper [K] in Maple, one obtains a system of primary invariants for H^{∗}(E)^{W} having degrees 4, 6, 7, and 8. Suitable choices for these invariants are the Dickson invariants d_{4}, d_{6}, d_{7}, described above, and d_{8}, the first Dickson invariant for H^{∗}((ℤ/2)^{4}),

d_{8} = w_{1}^{8} + w_{1}^{4}d_{4} + w_{1}^{2}d_{6} + w_{1}d_{7} + d_{4}^{2}.

The “folk-theorem” given by Kemper in [K] now tells us how many secondary invariants are necessary to generate the invariant ring. The number of secondary invariants is

\frac{(4 · 6 · 7 · 8)}{|W |} = 8.

To obtain the secondary invariants one determines, using Kemper’s package, the dimensions of the spaces of homogeneous invariants in degrees ≤ 13, which are given in the following table:

degree : 0 1 2 3 4 5 6 7 8 9 10 11 12 13
dimension : 1 0 0 0 1 0 1 1 3 1 2 2 5 3

Through degree 13, the polynomial algebra generated by the primary invariants has the following dimensions:

degree : 0 1 2 3 4 5 6 7 8 9 10 11 12 13
dimension : 1 0 0 0 1 0 1 1 2 0 1 1 3 1

Thus, the first degree in which there is a shortfall is 8. Therefore we must have a secondary invariant a_{8} of degree 8. Writing R for 𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}], we can compute the graded dimension in degrees 0 through 13 of the R-module R + R · a_{8} , which is of course an R-submodule of the invariant ring:

degree : 0 1 2 3 4 5 6 7 8 9 10 11 12 13
dimension : 1 0 0 0 1 0 1 1 3 0 1 1 4 1

We see now that the next shortfall is in dimension 9; therefore there must be a secondary invariant of degree 9, call it a_{9}. Again, we can compute the graded dimension of R + R · a_{8} + R · a_{9}, we obtain

degree : 0 1 2 3 4 5 6 7 8 9 10 11 12 13
dimension : 1 0 0 0 1 0 1 1 3 1 1 1 4 2

A comparison of this table with the table of dimensions of the full invariant ring, in just the manner of the previous two examples, shows that there must be secondary invariants in degrees 10, 11, 12, and 13, which we will call a_{10}, a_{11}, a_{12}, and a_{13}.

Finally we may check that W contains no “pseudo-reflection”, using Maple, and that we may therefore apply the Carlisle-Kropholler conjecture [B] to obtain the fact that the final secondary invariant must have degree 21. We call this final secondary invariant a_{21} and note that the computer tells us that we may in fact take a_{21} = a_{8} · a_{13}. Furthermore, using the package’s ability to compute Steenrod operations, we note that we may take a_{9} = Sq^{1}a_{8}, a_{11} = Sq^{2}a_{9}, a_{12} = Sq^{2}a_{10}, and a_{13} = Sq^{1}a_{12}.

Finally we note that this implies the Poincare series of the invariant ring is

\frac{1 + t^{8} + t^{9} + t^{10} + t^{11} + t^{12} + t^{13} + t^{21}}{(1 − t^{4})(1 − t^{6})(1 − t^{7})(1 − t^{8})}

and that the invariants are of the form

𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}](1, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{8}a_{13}).

If we restrict ourselves to the 2–Sylow subgroup H, then the corresponding Weyl group for E and F is the dihedral group D_{8}. In [AM2] it was shown that the cohomology H^{∗}(H, 𝔽_{2}) maps onto the D_{8} invariants in H^{∗}(E) and H^{∗}(F). Clearly the image of the restriction from H^{∗}(2A_{8}) will map into the L_{3}(2) invariants, however we in fact claim that it maps onto them in both cases. To show this we first recall a basic notion useful in cohomology calculations. A triple (Q, H, K) of groups with K ⊂ H ⊂ Q are said to form a weakly closed system if any subgroup in Q conjugate to K is already conjugate to K in H. The Cárdenas-Kuhn Theorem (see [AM1]) asserts that if (Q, K, V) is a weakly closed system with V = (ℤ/2)^{r} and [Q : K] odd, then the image of the restriction map from H^{∗}(Q) to H^{∗}(V) is precisely the image of the restriction from the cohomology of K intersected with the invariants under W_{Q}(V) = N_{G}(V)/V. In our case it is clear that E and F are both weakly closed in H ⊂ 2A_{8} (since they are the only (ℤ/2)^{4}’s in H), and so we deduce that in fact

Proposition 1.3:

The restriction homomorphism from H^{∗}(2A_{8}, 𝔽_{2}) to the cohomology of both E and F maps onto the L_{3}(2) invariants described above.

The Quillen-Venkov theorem (see [AM1]) shows that we have an exact sequence

0 → Rad (H^{∗}(2A_{8})) → H^{∗}(2A_{8}) ⟶^{res_{E}^{G}⊕res_{F}^{G}} H^{∗}(E)^{L_{3}(2)} ⊕ H^{∗}(F)^{L_{3}(2)} → M_{G} → 0

where the term on the left is the radical, i.e. the ideal of nilpotent elements in the cohomology while the term on the right is the quotient ring of double image classes – classes which must restrict non-trivially to both H^{∗}(E) and H^{∗}(F).

In what follows we will show that M_{G} ≅ 𝔽_{2}[d_{4}, d_{8}], whereas

Rad (H^{∗}(2A_{8})) ≅ 𝔽_{2}[d_{4}, d_{8}](u_{3}, u_{7}, u_{9}).

§2. DOUBLE IMAGE CLASSES

To continue the calculation of H^{∗}(G) we now determine the term

M_{G} = coker (res_{F}^{G} ⊕ res_{E}^{G})

which we can identify with “double image classes” in H^{∗}(E)⊕H^{∗}(F) following the notions in [AM2]. We begin by observing that H^{∗}(E)^{L_{3}(2)} will map onto M_{G}. Using this projection, we can endow the vector space M_{G} with a ring structure; using the other component H^{∗}(F)^{L_{3}(2)} we obtain the same structure, by symmetry. Next we relate the structure of M_{G} to the corresponding gadget for H = Syl_{2}(2A_{8}), which we denote by M_{H}. Using a stability argument it is easy to see that

Proposition 2.1:

The inclusion H ⊂ 2A_{8} induces a monomorphism of rings M_{G} → M_{H}.

We also need the following elementary lemma. Lemma 2.2:

Let V = E ∩ F, then the map H^{∗}(E)^{L_{3}(2)} ⊕ H^{∗}(F)^{L_{3}(2)} → H^{∗}(V) factors through M_{G}. Furthermore the ring structure on M_{G} is compatible with that in H^{∗}(V).

Now we determine the structure of the double image classes as follows.

Lemma 2.3:

The ring M_{G} = 𝔽_{2}[d_{4}, d_{8}].

Proof:

The ring M_{H} was computed in [AM2], and we have

M_{H} ≅ 𝔽_{2}[d_{4}, d_{8}](1, w_{3}, n_{7}).

Clearly d_{4} and d_{8} remain as double image classes for G using 2.2.

On the other hand, since there are no elements in M_{H} in dimensions 6 or 10, and the only element in M_{H} having dimension 8 is d_{8} it follows that d_{6} is not in the double image so there are classes d_{6}^{1} and d_{6}^{2} in H^{∗}(2A_{8}) which restrict to (d_{6}, 0), (0, d_{6}) respectively. Likewise, there must exist classes a_{8}^{1} and a_{8}^{2} in H^{∗}(2A_{8}) which restrict to (a_{8}, 0) and (0, a_{8}) respectively and a_{10}^{1}, a_{10}^{2} which restrict to (a_{10}, 0) and (0, a_{10}) respectively.

§3. DETERMINATION OF THE RADICAL

As was observed in [AM2], H contains S = Syl_{2}(L_{3}(4)) as an index 2 subgroup and the restriction map induces an injection of the radical Rad H^{∗}(H) → Rad H^{∗}(S) and hence an embedding Rad H^{∗}(2A_{8}) ⊂ Rad H^{∗}(S). We recall that

Rad H^{∗}(S) ≅ 𝔽_{2}[v_{4}, w_{4}](γ_{2}, β_{2}, γ_{3}, β_{3}, α_{5}).

The normalizer of S in G is an extension of the form S^{.}S_{3}, hence Rad H^{∗}(2A_{8}) ⊂ Rad H^{∗}(S)^{S_{3}}. We now determine these invariants. If we write S_{3} =< T, u >, where T has order 3 and u is an involution, then the explicit actions are given by (see [AM3])

T (v_{4}) = v_{4}, T (u_{4}) = u_{4}, T (γ_{2}) = β_{2}, T (β_{2}) = γ_{2} + β_{2}, T (β_{3}) = β_{3}, T (γ_{3}) = γ_{3}, T (α_{5}) = α_{5}

whereas

u(v_{4}) = w_{4}, u(w_{4}) = v_{4}, u(γ_{2}) = β_{2}, u(β_{2}) = γ_{2}, u(γ_{3}) = β_{3}, u(β_{3}) = γ_{3}, u(α_{5}) = α_{5}.

From this action we obtain the following invariants

Proposition 3.1:

(Rad H^{∗}(S))^{S_{3}} ≅ 𝔽_{2}[v_{4} + w_{4}, v_{4}w_{4}](γ_{3} + β_{3}, α_{5}, v_{4}γ_{3} + w_{4}β_{3}).

We can use the Lyndon–Hochschild–Serre spectral sequence for the extension

1 → ℤ/2 → 2A_{8} → A_{8} → 1

to analyze the cohomology of G in low dimensions. Using the results in [AMM], we in fact see that H^{i}(G) = 0 for i = 1, 2, 5, whereas H^{i}(G) ≅ 𝔽_{2} for i = 3, 4. We also have that H^{7}(G) ≅ H^{9}(G) ≅ (𝔽_{2})^{3} (details are provided in the next section). On the other hand, if as in [AM2] we choose a non-nilpotent k_{5} ∈ H^{5}(H) which restricts to α_{5}, then we may choose a class γ_{8} ∈ H^{8}(H) which restricts to v_{4}w_{4} and such that (γ_{8})^{i}k_{5} is nonnilpotent for all i ≥ 1. The details are as follows: the products γ_{8}^{i}k_{5} restrict non-trivially to the index 2 subgroup UT_{4}(2) (upper triangular 4×4 matrices over 𝔽_{2}), according to the explicit restriction data in [AM2]. However, the cohomology of this subgroup is detected on elementary abelian subgroups, from which non–nilpotence follows. Note that one can verify explicitly that res_{UT_{4}(2)}^{H}(γ_{4}k_{5}) = 0; indeed, following the notation in [AM2] this is a consequence of the fact that sγ_{4} = 0 and that s, k_{5} restrict non–trivially to precisely the same conjugacy classes of maximal detecting elementary abelian subgroups.

Now, by looking at the image of H^{∗}(G) in the cohomology of the maximal elementary abelian subgroups, we infer that there are at most two non–nilpotent seven and nine dimensional cohomology classes respectively in H^{∗}(G), and hence there must be a seven and a nine dimensional class in the radical. Combining these fact we conclude

Proposition 3.2:

Rad H^{∗}(2A_{8}) ≅ 𝔽_{2}[v_{4} + w_{4}, v_{4}w_{4}](γ_{3} + β_{3}, v_{4}γ_{3} + w_{4}β_{3}, α_{5}(v_{4} + w_{4})).

Summarizing, we have determined the cohomology of 2A_{8}:

Theorem 3.3:

Let G = 2A_{8}; then we have a short exact sequence

0 → 𝔽_{2}[d_{4}, d_{8}](u_{3}, u_{7}, u_{9}) → H^{∗}(G) ⟶^{res_{E}^{G}⊕res_{F}^{G}} H^{∗}(E)^{L_{3}(2)} ⊕ H^{∗}(F)^{L_{3}(2)} → 𝔽_{2}[d_{4}, d_{8}] → 0

where

H^{∗}(E)^{L_{3}(2)} = H^{∗}(F)^{L_{3}(2)} = 𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}](1, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{21}).

§4. THE SPECTRAL SEQUENCE OF THE CENTRAL EXTENSION

In this section we analyze the spectral sequence associated to the central extension

1 → ℤ/2 → G → A_{8} → 1.

This allows us, initially, to determine enough differentials in the spectral sequence to show that H^{7}(2A_{8}) ≅ H^{9}(2A_{8}) ≅ (𝔽_{2})^{3} as claimed in the proof of 3.1. Then, using 3.3 we complete the analyis of the spectral sequence obtaining very interesting interior differentials. To begin we recall the cohomology of the alternating group A_{8} [AMM]:

H^{∗}(A_{8}) ≅ 𝔽_{2}[σ_{2}, σ_{3}, c_{3}, σ_{4}, d_{6}, e_{6}, d_{7}, e_{7}](1, x_{5})/ < R >

where the subscript denotes the degree and R is the following set of relations:

d_{6}σ_{3} = e_{6}σ_{3} = σ_{3}(σ_{2}^{3} + σ_{2}σ_{4}) d_{6}e_{7} = d_{7}e_{6} = d_{7}e_{7} = 0 d_{7}σ_{2} = d_{7}σ_{3} = d_{7}c_{3} = d_{7}x_{5} = 0 e_{7}σ_{2} = e_{7}σ_{3} = e_{7}c_{3} = e_{7}x_{5} = 0 x_{5}σ_{3} = c_{3}σ_{3} = 0 d_{6}e_{6} = σ_{2}(σ_{2}c_{3}x_{5} + c_{3}^{2}σ_{4} + σ_{2}σ_{4}^{2} + σ_{2}^{5}) + x_{5}c_{3}σ_{4} x_{5}^{2} + x_{5}σ_{2}c_{3} + (d_{6} + e_{6})σ_{2}^{2} + σ_{4}c_{3}^{2} = 0

The following tables partially describe the action of the Steenrod algebra:

σ_{2}  σ_{3}  c_{3}
Sq^{1} σ_{3} + c_{3} 0  0
Sq^{2}  σ_{2}^{2}  σ_{2}σ_{3} σ_{2}c_{3} + x_{5}
Sq^{3}  0  σ_{3}^{2}  c_{3}^{2}

σ_{4}  x_{5}  d_{6}  d_{7}
Sq^{1}  x_{5}  0  d_{7} + σ_{3}(σ_{2}^{2} + σ_{4}) 0
Sq^{2}  σ_{2}σ_{4} + d_{6} + e_{6}  σ_{2}x_{5}  −  0
Sq^{3} d_{7} + e_{7} + c_{3}σ_{4} + σ_{3}σ_{4} + σ_{2}x_{5}  (c_{3} + σ_{3})x_{5}  −  0
Sq^{4}  σ_{4}^{2}  σ_{4}x_{5} + c_{3}(d_{6} + e_{6})  −  σ_{4}d_{7}
Sq^{5}  0  x_{5}^{2}  −  0
Sq^{6}  0  0  d_{6}^{2}  d_{6}d_{7}

The action of the Steenrod squares on e_{6}, e_{7} is analogous to that on d_{6}, d_{7} substituting the corresponding values.

We now consider the spectral sequence of the central extension. It has

E_{2}^{∗,∗} ≅ H^{∗}(A_{8}) ⊗ 𝔽_{2}[e],

where e represents the one dimensional generator on the fiber. We have that d^{2}(e) = σ_{2}. From the multiplicative relations above and a direct calculation using Macaulay to obtain the kernel of multiplication by σ_{2} in H^{∗}(A_{8}), we obtain the following E_{3}-term:

\left\{{𝔽_{2}[σ_{3} + c_{3},e_{6}, d_{6}](1, x_{5})/R′ ⊕ 𝔽_{2}[σ_{3} + c_{3}]σ_{3} \atop ⊕ 𝔽_{2}[d_{6}, d_{7}](d_{7}, d_{7}e) ⊕ 𝔽_{2}[e_{6}, e_{7}](e_{7}, e_{7}e)}\right\} ⊗ 𝔽_{2}[σ_{4}, e^{2}].

Here, R′ is the set of relations

\begin{aligned}x_{5}^{2} & = σ_{4}(c_{3} + σ_{3})c_{3}, \\ d_{6}e_{6} & = x_{5}(σ_{3} + c_{3})σ_{4}, \\ e_{6}σ_{3} & = 0, \\ d_{6}σ_{3} & = 0, \end{aligned}

together with the relations for the algebra in E_{2}. Note the two nontrivial permanent cocycles d_{7}e, e_{7}e in the E_{3}^{7,1}-term. They represent a_{8} and a′_{8} respectively.

Next we have d^{3}(e^{2}) = Sq^{1}σ_{2} = c_{3} + σ_{3}. Moreover, from the form of the E_{3}-term, we see that this is the only d^{3}-differential and E_{4} has the form

4.1  \left\{{𝔽_{2}[e_{6}, d_{6}](1, x_{5})/R′′⊕σ_{3} ⊕ 𝔽_{2}[d_{6}, d_{7}](d_{7}, ed_{7}, e^{2}d_{7}, e^{3}d_{7}) \atop ⊕ 𝔽_{2}[e_{6}, e_{7}](e_{7}, ee_{7}, e^{2}e_{7}, e^{3}e_{7})}\right\} ⊗ 𝔽_{2}[σ_{4}, e^{4}]

where R′′ is the pair of relations x_{5}^{2} = 0, d_{6}e_{6} = 0.

There are two possible d^{4}-differentials:

\begin{aligned}d^{4}(e^{3}d_{7}) & = ϵd_{6}x_{5} \\ d^{4}(e^{3}e_{7}) & = ϵ′e_{6}x_{5}, \end{aligned}

where ϵ, ϵ′ are 0 or 1. We will see later that these differentials actually must occur, but, in any case there can be no d^{4}-differential in total dimension < 10.

Now using Sq^{2} we see that d^{5}(e^{4}) = x_{5}. Again the relations imply that σ_{3}e^{4} and x_{5}e^{4} are non-trivial permanent cocycles in dimensions ≤ 9. Thus, in total degree seven the following classes survive to E_{∞}: d_{7}, e_{7}, σ_{3}e^{4}, whereas in total degree nine we have x_{5}e^{4}, d_{7}e^{2}, e_{7}e^{2}. This fills in the required information used in the computation of H^{∗}(G).

However we can now use 3.3 to determine the E_{∞} term in its totality, namely

Theorem 4.2:

In the spectral sequence for the extension 1 → ℤ/2 → 2A_{8} → A_{8} → 1 we have that there are two interior d^{4} differentials, d^{4}(e^{3}d_{7}) = d_{6}x_{5} and d^{4}(e^{3}d_{7}) = e_{6}x_{5}. The only remaining differential is d^{5}(e^{4}) = x_{5}, so E_{6} = E_{∞} is isomorphic to

\{(c_{3}, e^{4}c_{3}, e^{4}x_{5}) ⊕ \left[{𝔽_{2}[e_{6}]⊕𝔽_{2}[d_{6}]d_{6} ⊕ 𝔽_{2}[d_{6}, d_{7}](1, e, e^{2}, e^{3}d_{7})d_{7} \atop ⊕ 𝔽_{2}[e_{6}, e_{7}](1, e, e^{2}, e^{3}e_{6})e_{6}}\right] (1, e^{4})\} ⊗ 𝔽_{2}[σ_{4}, e^{8}].

Proof:

We need to consider d^{4} : E^{7,3} → E^{11,0}. If there is no d^{4} differential here, then 4.1 shows that E_{4} = E_{5} and, since d^{5}(e^{4}) = x_{5} it follows that E_{6} would equal E_{∞} and would have the form

\left\{{𝔽_{2}[e_{6}, d_{6}](1, e^{4}x_{5})/R′′⊕σ_{3}(1, e^{4}) ⊕ 𝔽_{2}[d_{6}, d_{7}](1, e)(1, e^{2})(1, e^{4})d_{7} \atop 𝔽_{2}[e_{6}e_{7}](1, e)(1, e^{2})(1, e^{4})e_{7}}\right\} ⊗ 𝔽_{2}[σ_{4}, e^{8}]

which is clearly inconsistent with the form of the answer previously determined. The only class available to be hit by d_{7}e^{3} is d_{6}x_{5}, and similarly the only class available to be hit by e_{7}e^{3} is e_{6}x_{5}. It follows that these differentials must be non-trivial and E_{5} has the form

\left\{{(c_{3}, x_{5}) ⊕ 𝔽_{2}[e_{6}] ⊕ 𝔽_{2}[d_{6}]d_{6} ⊕ 𝔽_{2}[d_{6}, d_{7}](1, e, e^{2}, e^{3}d_{7})d_{7} \atop 𝔽_{2}[e_{6}, d_{7}](1, e, e^{2}, e^{3}e_{7})e_{7}}\right\} ⊗ 𝔽_{2}[σ_{4}, e^{4}].

The d^{5} differential on e^{4} now implies that E_{6} has the form given in 4.2 which has the same Poincaré series as the answer obtained in §3.

§5. THE SYLOW SUBGROUP OF Ly AND THE COHOMOLOGY OF 2S_{8}

The Sylow subgroup of Ly – which is also Syl_{2}(2A_{10}) – can be given as an extension in many ways. For example, as a split extension it has the form Syl_{2}(J_{2}): 2 and Syl_{2}(McL): 2, while as a non-split central extension it is given as

2 ◁ Syl_{2}(Ly)⟶^{π}2 ≀ 2 ≀ 2 = Syl_{2}(S_{8}).

A convenient description is as the extension Syl_{2}(L_{3}(4)): 2^{2} where Syl_{2}(L_{3}(4)) is the 3×3upper triangular matrices with entries in 𝔽_{4}:

Syl_{2}(L_{3}(4)) = UT_{3}(4) = ⟨( \begin{matrix}1 & α & β \\ 0 & 1 & γ \\ 0 & 0 & 1 \end{matrix} ) | α, β, γ ∈ 𝔽_{4}⟩

and 2^{2} is generated by the two automorphisms of Syl_{2}(L_{3}(4)), Galois conjugation, g,

( \begin{matrix}1 & α & β \\ 0 & 1 & γ \\ 0 & 0 & 1 \end{matrix} )^{g} = ( \begin{matrix}1 & α^{2} & β^{2} \\ 0 & 1 & γ^{2} \\ 0 & 0 & 1 \end{matrix} )

and the automorphism corresponding to the automorphism of L_{3}(4) given by M ↔ M^{−t} for each M ∈ GL_{3}(𝔽_{4}):

( \begin{matrix}1 & α & β \\ 0 & 1 & γ \\ 0 & 0 & 1 \end{matrix} )^{A} = ( \begin{matrix}1 & γ & β + αγ \\ 0 & 1 & α \\ 0 & 0 & 1 \end{matrix} ) .

Note that A is really the composite

m ↦ ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) m^{−t} ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} )

and thus extends to an automorphism of all of GL_{3}(𝔽_{4}) leaving the subgroup of upper triangular matrices invariant. Note also that UT_{3}(4) contains exactly two copies of 2^{4}: the subgroups

2_{I}^{4} = ⟨( \begin{matrix}1 & α & β \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )⟩ , 2_{II}^{4} = ⟨( \begin{matrix}1 & 0 & α \\ 0 & 1 & β \\ 0 & 0 & 1 \end{matrix} )⟩ .

We also denote the generators of the center of UT_{3}(4) as

T = ( \begin{matrix}1 & 0 & ζ_{3} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , Z = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Note that the center of Syl_{2}(Ly) = UT_{3}(4): ⟨g, A⟩ is the single ℤ/2 = ⟨Z⟩. The subgroup UT_{3}(4): ⟨g⟩ = Syl_{2}(McL) and is also Syl_{2}(2A_{8}). It has center ⟨Z⟩ and exactly two subgroups isomorphic to 2^{4}, 2_{I}^{4}, and 2_{II}^{4}, both of which are normal.

The subgroup UT_{3}(4): ⟨gA⟩ = Syl_{2}(J_{2}) = Syl_{2}(J_{3}) and here the two 2^{4}’s are fused. Finally, in the full group, Syl_{2}(Ly) = Syl_{2}(2S_{8}) = Syl_{2}(2A_{10}), there are again only the two 2^{4} subgroups above, but they are conjugate. However, there are a number of new 2^{3}’s, for example ⟨g, A, Z⟩ and ⟨A, T, Z⟩. In all three of the groups 2S_{8} ⊂ 2A_{10} ⊂ Ly there are precisely two conjugacy classes of maximal elementary abelian subgroups, and we can take as representatives the 2^{4} above and the 2^{3} defined by ⟨A, T, Z⟩. This of course immediately implies that both of these elementary abelian groups form part of weakly closed systems in 2S_{8} ⊂ 2A_{10} and 2S_{8} ⊂ Ly.

Now we consider the extension 2A_{8}: 2 = 2S_{8} ⊂ 2A_{10}. The action of the extending ℤ/2 exchanges the two 2^{4}’s, consequently, on passing to cohomology it exchanges the generators corresponding to (d_{6}, 0), (0, d_{6}), (d_{7}, 0), (0, d_{7}) and the similar elements corresponding to the terms (a_{i}, 0), (0, a_{i}), while it and fixes the remaining generators. It follows that the E_{2}-term of the resulting Lyndon-Hochschild-Serre spectral sequence is

𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}](1, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{21}) ⨁ 𝔽_{2}[d_{4}, d_{8}, w](w, u_{3}, u_{7}, u_{9})

where w represents the cohomology class dual to the extending ℤ/2.

Lemma 5.1:

There are two conjugacy classes of maximal elementary 2-groups in 2S_{8}, 2A_{10}, and at most two in Ly, the 2^{4} discussed above which is weakly closed in Syl_{2}(2S_{8}) ⊂ 2S_{8}, and one of the new 2^{3}’s – the choice we will take in the following is ⟨A, T, Z⟩ – which is weakly closed in 2S_{8} ⊂ 2A_{10}. The Weyl groups of ⟨A, T, Z⟩ in 2S_{8} and 2A_{10} are D_{8} and S_{4} respectively.

Proof: We start by verifying the following claim.

Claim: The maximal elementary abelian 2-groups in 2A_{10}, 2S_{8} are in one to one correspondence (via the quotient maps) with the elementary abelian 2-groups in A_{10}, S_{8} whose nonidentity elements are all products of four disjoint transpositions and are maximal with respect to this condition.

To see this, one should note that 2A_{5} has no non-central involutions, and both A_{8} and A_{10} have two conjugacy classes of involutions, one which lifts to an involution while the other lifts to an element of order four [Co]. We also have the commuting diagram

2A_{5} ↪ 2A_{8} ↪ 2A_{10}
↓ ↓ ↓
A_{5} ↪ A_{8} ↪ A_{10}

which tells us that we must have the standard embedding of A_{5} in A_{10}. Since the noncentral involutions of A_{10} must project onto involutions not contained in A_{5}, they must have cycle type (1, 2)(3, 4)(5, 6)(7, 8).

If we consider S_{8} ⊂ A_{10} generated by (1, 2)(9, 10), (2, 3)(9, 10), ... (7, 8)(9, 10), then we see that the remarks above apply to S_{8} as well.

Claim: The maximal elementary abelian 2-groups in S_{n} are (up to conjugacy) of the form V_{1}^{i_{1}} × ⋯ × V_{r}^{i_{r}}, where V_{j} ≅ 2^{j} and the permutation action is the regular representation. Here, the number of letters permuted by the subgroup is 2i_{1} + ⋯ + 2^{r}i_{r}, and this number is either n or n − 1 (see e.g., [AM1], Chapter VI).

Observe that the subgroups V_{i} described above are contained in the alternating group if i ≥ 2. Furthermore, note that the subgroup E_{r} of even permutations contained in V_{1}^{r} is not maximal in A_{2r} if r = 2, since the even permutations in V_{1} × V_{1} are strictly contained in V_{2}. On the other hand if r ≥ 3 then E_{r} is a maximal elementary abelian 2-group in A_{2r}, and is the unique such group with r 2-element orbits.

Altering the arguments for the symmetric group to reflect these remarks we obtain the following:

Proposition 5.2:

Every maximal elementary abelian 2-group in A_{n} is conjugate to one of the form E_{i_{1}}×V_{2}^{i_{2}}×⋯×V_{r}^{i_{r}}, where V_{j} ≈ 2^{j} and the permutation action is the regular representation of 2^{j}. Here, the number of letters permuted by the subgroup is 2i_{1} + 4i_{2} + ⋯ + 2^{r}i_{r}, and this number is n, n − 1, n − 2, or n − 3.

To complete the classification of the conjugacy classes of elementary abelian 2-groups in A_{n}, we need the following remark. Remark: Suppose that 8n = 2^{3}i_{1} + 2^{4}i_{4} + ⋯ + 2^{r}i_{r}. Then a conjugacy class of maximal elementary abelian 2-groups in S_{8n} or S_{8n+1} of the form V_{3}^{i_{3}} × V_{4}^{i_{4}} × ⋯ × V_{r}^{i_{r}} (which is of course also maximal in A_{8n}, A_{8n+1}, A_{8n+2}, or A_{8n+3}) splits into two conjugacy classes of elementary abelian 2-groups in A_{8n} or A_{8n+1}. The other types of maximal elementary abelian 2-groups in A_{m} are conjugate if and only if they have the same form.

The essential point in the proof of this remark is the fact that the number of conjugates of a subgroup E ⊂ G is the index [G : N_{G}(E)] of the normalizer. Applying this to the case E ⊂ A_{m}, we see that the number of conjugates of a E in S_{m} is either the same as the number of conjugates of E in A_{m}, or twice as many, depending on whether there is a transposition of S_{m} normalizing E. Having made this point, our remark reduces to the fact that if E = V_{3}^{i_{3}} × V_{4}^{i_{4}} × ⋯ × V_{r}^{i_{r}}, there is no transposition of S_{8n} (or S_{8n+1}) normalizing E, and that there is one in the other cases.

From what we have shown it follows that (up to conjugacy) the maximal elementary abelian 2-groups in S_{8} are V_{3}, V_{2}^{2}, V_{2} × V_{1}^{2}, and V_{1}^{4}. The conjugacy classes of maximal elementary abelian subgroups of A_{10} are represented by V_{3}, V_{2}^{2}, V_{2} × E_{3}, and E_{5}.

Looking at the subgroups of the groups from the lists above, the only ones whose elements satisfy the necessary condition on cycle type are the conjugacy classes of

Ṽ_{3} =< (1, 2)(3, 4)(5, 6)(7, 8), (1, 3)(2, 4)(5, 7)(6, 8), (1, 5)(2, 6)(3, 7)(4, 8) >

and

M̃_{3} =< (1, 3)(2, 4)(5, 6)(9, 10), (1, 4)(2, 3)(7, 8)(9, 10) > .

Then M_{3} and V_{3}, the lifts of M̃_{3} and Ṽ_{3}, represent the maximal elementary abelian 2groups of both 2A_{10} and 2S_{8}.

Finally, we need to check the Weyl groups of M_{3} in 2S_{8} and 2A_{10} are as stated. Clearly, V_{2} = ⟨(1, 3)(2, 4), (1, 4)(2, 3)⟩ and (1, 2)(5, 7)(6, 8) normalize this group, and these together generate a D_{8}. But since (9, 10) must be fixed by the elements of 2S_{8} it follows that there can be no element of order three which normalizes it in 2S_{8}. But the element of order three in 2A_{10} which cyclically permutes the transpositions (5, 6), (7, 8), and (9, 10) while simultaneously taking (1, 3)(2, 4) to (1, 2)(3, 4) also normalizes M_{3} and the proof of 5.1 is complete.

We have the following calculation of the invariant subring for ⟨A, T, Z⟩ under the action of the D_{8}-Weyl group for 2S_{8}:

Lemma 5.3:

𝔽_{2}[w, t, z]^{D_{8}} = 𝔽_{2}[w, t(t + w), d_{4}] where w is dual to the extending element A = (1, 2)(5, 6)(7, 8), t is dual to (1, 2)(3, 4)(5, 8)(6, 7) corresponding to T, and z is dual to the central element, Z. Also,

d_{4} = z^{4} + z^{2}d_{2} + zd_{3} + d_{2}^{2}

is the generating element of the Dickson algebra, with d_{2} = w^{2} + tw + t^{2}, d_{3} = wt(w + t).

Proof:

We start with the action of the element of order two which has the form z ↦ z + t, t ↦ t, w ↦ w. This give the invariant subring

𝔽_{2}[w, t, z(z + t)].

Now consider the action of the second element of order two fixing t, w, z ↦ z + w. Then z(z + t) ↦ (z + w)(z + t + w) = z(z + t) + w(t + w) and the invariant subring is clearly 𝔽_{2}[w, t, z(z +t)(z +w)(z +t+w)]. Expanding out this last term is z^{4} +z^{2}d_{2} +zd_{3}. Finally, the last element fixes w, z and takes t to t + w. Applying this, we clearly get the asserted ring of invariants.

Now here is the main result of this section.

Theorem 5.4:

In the spectral sequence above for H^{∗}(2S_{8}) we have that E_{2} = E_{∞}. Moreover, the restriction map H^{∗}(2S_{8})⟶H^{∗}(2^{4})^{L_{3}(2)} ⊕ H^{∗}(2^{3})^{D_{8}} is injective in cohomology. Moreover, the quotient by the image has the form 𝔽_{2}[d_{4}, d_{8}] with d_{4} being the image of d_{4} from H^{∗}(2^{4}) and (w^{2} + t(w + t))^{2} from H^{∗}(2^{3}) while d_{8} is the image of d_{8} and d_{4}^{2}, respectively.

Proof:

We have ([AM1] p. 212) that H^{∗}(S_{8}) = 𝔽_{2}[σ_{1}, σ_{2}, σ_{3}, σ_{4}, c_{3}, d_{6}, d_{7}](x_{5})/R where R is the set of relations

d_{6}σ_{1} = d_{6}σ_{3} = 0
d_{7}σ_{1} = d_{7}σ_{2} = d_{7}σ_{3} = d_{7}c_{3} = d_{7}x_{5} = 0
x_{5}σ_{3} + c_{3}σ_{4}σ_{1} = 0
c_{3}(σ_{3} + σ_{1}σ_{2}) + σ_{1}x_{5} = 0
x_{5}^{2} + x_{5}σ_{2}c_{3} + d_{6}σ_{2}^{2} + σ_{4}c_{3}^{2} = 0.

We also have the commutative diagram

ℤ/2 ⟶^{=} ℤ/2 ⟶^{=} ℤ/2
↓  ↓  ↓
2^{3}  ↪ 2^{3} · 2^{2} ↪ 2S_{8}
↓  ↓  ↓
W = 2^{2} ⟶^{Δ} V_{1}^{2} × K ↪ S_{8},

where Δ is the diagonal map, (identifying both V_{1}^{2} and K with 2^{2} and noting that the generators of W go diagonally to generators of the first and second 2^{2}’s). From this it is direct to determine the image of the restriction map from the generators of H^{∗}(S_{8}) to H^{∗}(2^{3}) as follows

\{ \begin{matrix}σ_{1} ↦ σ_{1} ⊗ 1 ↦ w \\ σ_{3} ↦ σ_{1} ⊗ d_{2} ↦ w^{3} + d_{3} \\ c_{3} ↦ 1 ⊗ d_{3} ↦ d_{3} \\ σ_{4} ↦ σ_{2} ⊗ d_{2} ↦ t(t + w)d_{2} \\ x_{5} ↦ σ_{2} ⊗ d_{3} ↦ t(t + w)d_{3} \\ d_{6} ↦ 0 \\ d_{7} ↦ 0. \end{matrix}

This shows that the subalgebra 𝔽_{2}[w, d_{2}^{2}](1, d_{3}) in H^{∗}(2^{3}) is contained in the image from the cohomology of H^{∗}(2S_{8}) as the subring coming from the composition

H^{∗}(S_{8})⟶H^{∗}(2S_{8})⟶^{res}H^{∗}(2^{3}).

Also, by the argument of [AM3], §6, there is a real representation of Ly so that the Stiefel-Whitney class w_{8} restricts to d_{8} in H^{∗}(2^{4}) and d_{4}^{2} ∈ H^{∗}(⟨Z, W ⟩). It follows that the image of res: H^{∗}(2S_{8})⟶H^{∗}(⟨ZW ⟩) contains the subalgebra 𝔽_{2}[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}).

We now use the Lyndon-Hochschild-Serre spectral sequences of the central extensions 2 ◁ 2^{3} ⟶^{π}W and 2 ◁ 2S_{8} ⟶^{π}S_{8} together with naturality to show that the remaining classes a_{7} and X_{9} also restrict non-trivially to H^{∗}(⟨Z, W ⟩).

In the spectral sequence for 2S_{8} we write E_{2} = H^{∗}(S_{8}) ⊗ 𝔽_{2}[z], while in the spectral sequence for 2^{3} we have

\begin{aligned}E_{2} & = H^{∗}(W) ⊗ 𝔽_{2}[z] \\ & = 𝔽_{2}[w, t] ⊗ 𝔽_{2}[z]. \end{aligned}

Moreover, the restriction map res: E_{2}(2S_{8})⟶E_{2}(2^{3}) has the form res ⊗ id, and is natural with respect to differentials. Of course, the differentials in the spectral sequence for 2^{3} are all trivial, which implies that no element in the spectral sequence for H^{∗}(2S_{8}) which restricts non-trivially to the sequence for 2^{3} can be in the image of a differential in the spectral sequence for 2S_{8}.

The differential d^{2} is given by d^{2}(z) = σ_{2} + σ_{1}^{2} while the basic higher differentials are given by

\begin{aligned}d^{3}(z^{2}) & = σ_{1}σ_{2} + σ_{3} + c_{3} \\ d^{5}(z^{4}) & = x_{5} + c_{3}σ_{1}^{2} + σ_{1}σ_{4}. \end{aligned}

An easy calculation shows that d^{5}(x_{5}z^{4}) ∈ H^{10}(S_{8}) is in the image of the differentials d^{2} and d^{3}, and a similar calculation shows the same thing for d^{5}(c_{3}z^{4}). Consequently, since these classes restrict non-trivially to the spectral sequence for H^{∗}(2^{3}), it follows that they survive to E_{∞} and represent non-trivial classes in H^{∗}(2S_{8}).

But the argument above actually shows more. It shows that, up to filtration, the class represented by x_{5}z^{4} restricts to res(x_{5})z^{4}, i.e., the image of restriction on this class has the form

t(t + w)d_{3}z^{4} + v_{6}(t, w)z^{3} + v_{7}(t, w)z^{2} + v_{8}(t, w)z + v_{9}(t, w).

On the other hand the image of this class lies in the invariant subring, so it must have the form t(t+w)d_{3}d_{4} +v_{9}(t, w) with v_{9}(t, w) also in the invariant subring. However, it is direct to check that every element in dimension nine in 𝔽_{2}[w, t(t + w)] is already in the image of restriction from H^{∗}(S_{8}), and t(t + w)d_{3}d_{4} is in the image of restriction from H^{∗}(2S_{8}).

A similar argument with c_{3}z^{4} shows that d_{3}d_{4} is in the image of restriction from H^{∗}(2S_{8}). But the smallest sub-ring of H^{∗}(2^{3}) containing these elements and res(H^{∗}(S_{8})) has the form

𝔽_{2}[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}).

This completes the proof.

Corollary 5.5:

The group ⟨A, T, Z⟩ ≅ M_{3} is weakly closed in 2S_{8} ⊂ Ly. Also, the Weyl groups of ⟨A, T, Z⟩ in Ly is L_{3}(2).

Proof: Note that from the work on 2S_{8} we already know that H^{∗}(Ly) is detected by restriction to H^{∗}(2^{4})^{A_{7}} ⊕ H^{∗}(2^{3}). If the 2^{3} fused into 2^{4} then H^{∗}(Ly) would be detected by H^{∗}(2^{4})^{A_{7}} alone. But from [AM3], §6, we have

Proposition 5.6:

There is a real representation of Ly so that the Stiefel-Whitney class w_{8} restricts to d_{8} in H^{∗}(2^{4}) and to d_{4}^{2} ∈ H^{∗}(⟨A, T, Z⟩).

Since there are no other invariants in dimensions 7 or 9 it follows that d_{8} must be the restriction of a torsion free cohomology class and this is impossible. Consequently the 2^{3} cannot fuse into the 2^{4}.

Similarly, we show that the Weyl group of this 2^{3} is L_{3}(2). Indeed, since the part of the proof of 5.1 already finished shows that the Weyl group is at least S_{4} which is maximal in L_{3}(2), it follows that the Weyl group is either S_{4} or L_{3}(2). But if it were S_{4} then weak closure would imply that the restriction to H^{∗}(2^{3}) is the same as the restriction of H^{∗}(2A_{10}). On the other hand both d_{3} and d_{2}^{2} are in this image, but d_{2}^{2} is a double image class, and hence d_{4} would have to be in the invariant subring H^{∗}(2^{4})^{A_{7}}. This is impossible, so it follows that the Weyl group is L_{3}(2).

Summarizing we have the following result which reduces the calculation of H^{∗}(2A_{10}) and H^{∗}(Ly) to direct calculations with rings of invariants.

Corollary 5.7:

H^{∗}(2A_{10}) and H^{∗}(Ly) are both detected by restriction to the direct sum H^{∗}(2^{4}) ⊕ H^{∗}(⟨A, T, Z⟩). In particular, since

⟨Z, A, T ⟩ = ⟨Z, W ⟩ ⊂ 2S_{8} ⊂ \{ \begin{matrix}2A_{10} \\ Ly \end{matrix}

are both weakly closed systems it follows that the image of restriction from H^{∗}(2A_{10}) to H^{∗}(⟨Z, W ⟩) is given as the intersection

𝔽_{2}[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}) ∩ H^{∗}(⟨Z, W ⟩)^{S_{4}}

while the image of restriction from H^{∗}(Ly) is given as

𝔽_{2}[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}) ∩ 𝔽_{2}[d_{4}, d_{6}, d_{7}].

§6. THE COHOMOLOGY OF 2A_{10}

We will now make explicit the cohomology of 2A_{10} using the results in §5. First we recall that the S_{4}–invariants can be computed from the usual symmetric invariants and in our notation can be expressed as

𝔽_{2}[w, t, z]^{S_{4}} = 𝔽_{2}[d_{2}, d_{3}, d_{4}].

Hence we immediately obtain the classes d_{3}, d_{2}^{2}, d_{3}d_{4}, and d_{4}^{2} in the intersection. Moreover note the relations

w^{2}d_{3} + wd_{2}^{2} + w^{5} = d_{2}d_{3}, w^{2} + t(t + w)d_{3}d_{4} = d_{2}d_{3}d_{4}.

From this we see that the classes d_{2}d_{3} and d_{2}d_{3}d_{4} are also in the intersection, and hence it is at least as big as

𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}).

This is in fact the totality of the intersection; complete details for this computation are provided in an appendix. We can describe the cohomology of 2A_{10} as follows:

Theorem 6.1:

The cohomology of 2A_{10} is described by a short exact detection sequence of the form

0 → H^{∗}(2A_{10}) → H^{∗}((ℤ/2)^{4})^{L_{3}(2)} ⊕ 𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}) → 𝔽_{2}[h_{4}, h_{8}] → 0

where h_{4}, h_{8} correspond to the double image classes (d_{4}, d_{2}^{2}) and (d_{8}, d_{4}^{2}).

§7. THE COHOMOLOGY OF THE LYONS GROUP

Now we turn to the determination of the cohomology of the sporadic group Ly. From our previous results it will be completely determined once we have

𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}) ∩ 𝔽_{2}[d_{4}, d_{6}, d_{7}].

Lemma 7.1:

The intersection above is given as

𝔽_{2}[d_{4}^{2}, d_{6}^{2}, d_{7}](1, d_{4}d_{7}, d_{6}d_{7}, d_{4}d_{6}d_{7}) = 𝔽_{2}[d_{4}^{2}, d_{6}^{2}] ⊕ 𝔽_{2}[d_{4}, d_{6}, d_{7}]d_{7}.

Proof:

We can write the ring

𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}) = 𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}) ⊕ 𝔽_{2}[d_{2}, d_{3}, d_{4}^{2}]d_{3}d_{4}.

On the other hand we have the decompositions

\begin{aligned}d_{6} & = d_{2}d_{4} + d_{2}^{3} + d_{3}^{2}, \\ d_{7} & = d_{3}d_{4} + d_{2}^{2}d_{3}, \end{aligned}

so we can modifiy the decomposition above to the new decomposition

𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}) ⊕ 𝔽_{2}[d_{2}, d_{3}, d_{4}^{2}]d_{7}.

But we can further write the first summand in the form

𝔽_{2}[d_{2}^{2}, d_{3}^{2}, d_{4}^{2}](1, d_{3}, d_{2}d_{3}, d_{2}d_{3}^{2}).

Using this we can prove the

Sublemma 7.2:

The intersection of the ring above with 𝔽_{2}[d_{4}, d_{6}] is 𝔽_{2}[d_{4}^{2}, d_{6}^{2}]. Proof:

Clearly, the intersection is 𝔽_{2}[d_{4}, d_{6}] ∩ 𝔽_{2}[d_{2}^{2}, d_{3}^{2}, d_{4}^{2}](1, d_{3}, d_{2}d_{3}, d_{2}d_{3}^{2}). But to get a term involving d_{6}^{i} we must have a term involving d_{2}^{i}d_{4}^{i}. Moreover, it cannot have any d_{3}’s involved. Hence the intersection must lie in 𝔽_{2}[d_{2}^{2}, d_{3}^{2}, d_{4}^{2}]. But this intersection with 𝔽_{2}[d_{4}, d_{6}, d_{7}] is 𝔽_{2}[d_{4}^{2}, d_{6}^{2}, d_{7}^{2}].

Note that we can replace 𝔽_{2}[d_{2}^{2}, d_{3}, d_{4}^{2}](d_{3}d_{4}^{2}, d_{2}d_{3}d_{4}^{2}) by 𝔽_{2}[d_{2}^{2}, d_{3}^{2}, d_{4}^{2}]d_{7}d_{4}, and this summed with 𝔽_{2}[d_{2}, d_{3}, d_{4}^{2}]d_{7} is 𝔽_{2}[d_{2}, d_{3}, d_{4}]d_{7}. It follows that the intersection contains 𝔽_{2}[d_{4}, d_{6}, d_{7}]d_{7} and since the quotient of 𝔽_{2}[d_{4}, d_{6}, d_{7}] by this ideal is 𝔽_{2}[d_{4}, d_{6}] it follows that the intersection is given as

𝔽_{2}[d_{4}, d_{6}] ∩ 𝔽_{2}[d_{3}, d_{2}^{2}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4})

direct summed with the ideal (d_{7}).

Now we are able to describe the cohomology of Ly. Recall that the Weyl group of the 2^{4} in Ly is A_{7}, and that the ring of A_{7}-invariants has been determined in [AM4] as

H^{∗}(2^{4})^{A_{7}} = 𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}](1, x_{18}, x_{20}, x_{21}, x_{24}, x_{25}, x_{27}, x_{45}).

Theorem 7.3:

The cohomology of Ly is given by the short exact detection sequence

0⟶H^{∗}(Ly)⟶H^{∗}(2^{4})^{A_{7}} ⊕ 𝔽_{2}[d_{4}^{2}, d_{6}^{2}, d_{7}](1, d_{4}d_{7}, d_{6}d_{7}, d_{4}d_{6}d_{7})⟶𝔽_{2}[h_{8}, h_{12}]⟶0

where h_{8} and h_{12} correspond to the double image classes (d_{8}, d_{4}^{2}) and (d_{12}, d_{6}^{2}).

Remark: We would like to point out that from the spectral sequences for computing H^{∗}(2A_{10}) and H^{∗}(2S_{8}) it is apparent that their cohomology rings have depth equal to three i.e. there is a regular sequence of length three. On the other hand, an explicit computer algebra verification shows that the centralizers of the (ℤ/2)^{3} subgroups in these groups are either the (ℤ/2)^{3} themselves or a (ℤ/2)^{4}. A theorem due to Carlson [C] indicates that a depth three cohomology ring must be detected on the centralizers of rank three elementary abelian subgroups. Of course, currently, the only way to verify the conditions of Carlson’s result is to complete the calculations of the cohomology groups since differentials in the spectral sequences can lower ranks. But it would be very interesting if one could find independent methods for showing that these conditions are satisfied.

APPENDIX

Let k be the field of two elements and k[d_{4}, d_{6}, d_{7}] the Dickson algebra of GL_{3}(k)- invariants in the polynomial ring k[w, t, z]. Let d_{2}, d_{3} be the generators of the GL_{2}(k)- invariants in k[w, t]. In this appendix we will show that

k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}) ∩ k[d_{2}, d_{3}, d_{4}] = k[d_{2}^{2}, d_{3}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}).

To begin we observe that since d_{2} = w^{2} + t(t + w), it must follow that

k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}) = k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}).

We will require the following

Lemma A1

The subalgebra k[w, d_{2}, d_{3}] of k[w, t] is a free k[d_{2}, d_{3}] -module with basis {1, w, w^{2}}. Proof: This is a special case of a more general fact about polynomial invariants (see [Mit]). However, for the sake of completeness we indicate a proof here: first, from the usual lemma on integral extensions, we have that {1, w, w^{2}} generates k[w, d_{2}, d_{3}] as a k[d_{2}, d_{3}]-module. So it is enough to prove that there is no relation p + qw + rw^{2} = 0 with p, q, r ∈ k[d_{2}, d_{3}]. Such a relation would be an equation in the polynomial ring k[w, t] and therefore remains true after applying an automorphism of that ring. So, choosing φ, θ ∈ GL_{2}(k) such that φ(w) = t, θ(w) = w + t, we have the relations p + qt + wt^{2} = 0 and p + q(w + t) + r(w^{2} + t^{2}) = 0. Adding gives p = 0. Since k[w, t] is a domain, it follows from qw + rw^{2} = 0 that q + rw = 0. The same argument gives q = 0 and it then follows that r = 0.

It follows from the lemma that k[d_{2}, d_{3}, d_{4}](1, w, w^{2}) is a free k[d_{2}, d_{3}, d_{4}]-module with basis {1, w, w^{2}}. Since k[d_{2}, d_{3}, d_{4}] is a free module over k[d_{2}^{2}, d_{3}, d_{4}^{2}] with basis {1, d_{2}, d_{4}, d_{2}d_{4}}, it follows that: k[d_{2}, d_{3}, d_{4}](1, w, w^{2}) is a free module over k[d_{2}^{2}, d_{3}, d_{4}^{2}] with basis the twelve elements obtained from the product {1, w, w^{2}} · {1, d_{2}, d_{4}, d_{2}d_{4}}, i.e.

1, d_{2}, d_{4}, d_{2}d_{4}, w, wd_{2}, wd_{4}, wd_{2}d_{4}, w^{2}, w^{2}d_{2}, w^{2}d_{4}, w^{2}d_{2}d_{4}.

We will call the free R-module mentioned above M. We use the fact that if A ⊂ B ⊂ C are rings and C is a finitely-generated free B-module with basis {c_{i}} and B is a finitelygenerated free A-module with basis {b_{j}}, then C is a finitely-generated free A-module with basis {b_{j}c_{i}}.

In what follows we will need the following formal lemma on the same theme.

Lemma A2:

If α, β are integral elements over R and {α_{1}, …, α_{n}} generate R[α] as an R-module, while {β_{1}, …, β_{m}} generate R[β] as an R-module, then {α_{i}β_{j}} is a set of R-module generators for R[α, β].

The proof follows from regarding R[α, β] as R[α][β], which makes sense since β integral over R implies β integral over R[α]. We note that the system of generators in this lemma may be very redundant.

The Intersection

Let R = k[d_{2}^{2}, d_{3}, d_{4}^{2}]. It follows from the remarks in the previous section that w is integral over R. In fact, X^{6} + d_{2}^{2}X^{2} + d_{3}^{2} = 0 is an integral equation for w over R. It follows that k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}) is an integral extension of R with {1, w, …, w^{5}} as a set of R-module generators. Furthermore d_{3}d_{4}, d_{2}d_{3}d_{4} are integral over R, with equations X^{2} + d_{3}^{2}d_{4}^{2} and X^{2} + d_{2}^{2}d_{3}^{2}d_{4}^{2} repectively.

It follows from Lemma A2 that the 24 elements given by all triple products from the set {1, w, …, w^{5}} · {1, d_{3}d_{4}} · {1, d_{2}d_{3}d_{4}} generate

k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4})

as an R-module. We denote k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}) by U when regarding it as an R-module. When k[d_{2}, d_{3}, d_{4}] is regarded as an R-module, it will be denoted by V. Notice that U and V are R-submodules of M. In the notation just described, the purpose of this note is to find generators for U ∩ V.

To begin describing a simpler generating set for the R-module U, let us first examine the subset {1, w, …, w^{5}} of the generators of U. Since w^{3} + d_{2}w + d_{3} = 0 and d_{3} ∈ R, the R-submodule of U generated by {1, w, w^{2}, w^{3}} is the same as that generated by {1, w, w^{2}, d_{2}w}. Similar reasoning, plus the equation w^{5} + d_{2}^{2}w + d_{2}d_{3} = 0, gives that the R-submodule generated by {1, w, …, w^{5}} is the same as that generated by {1, w, w^{2}, wd_{2}, w^{2}d_{2}, d_{2}d_{3}}.

Thus we can replace the first set in the 24 generators remark above with the set {1, w, w^{2}, wd_{2}, w^{2}d_{2}, d_{2}d_{3}}.

As U is a submodule of the free module M defined in the previous section we can write our new generating set in terms of the basis previously described from the free module M. To make it clear when we are regarding a polynomial as one of the free generators of M, we will represent the free generators in boldface. Thus M is the free R-module on generators

1, d_{2}, d_{4}, d_{2}d_{4}, w, wd_{2}, wd_{4}, wd_{2}d_{4}, w^{2}, w^{2}d_{2}, w^{2}d_{4}, w^{2}d_{2}d_{4}.

Multiplying out the 24 triple products yields the following generators of U (regarded here as elements of M):

1, w, w^{2}, wd_{2}, w^{2}d_{2}, d_{3}d_{2}, d_{3}d_{4}, d_{3}wd_{4}, d_{3}w^{2}d_{4}, d_{3}wd_{2}d_{4}, d_{3}w^{2}d_{2}d_{4},

d_{3}d_{2}d_{4}, d_{3}d_{2}d_{4}, d_{3}wd_{2}d_{4}, d_{3}w^{2}d_{2}d_{4}, d_{2}^{2}d_{3}wd_{4}, d_{2}^{2}d_{3}w^{2}d_{4}, d_{2}^{2}d_{3}^{2}d_{4},

d_{3}^{2}d_{4}^{2}d_{2}, d_{3}^{2}d_{4}^{2}d_{2}w, d_{3}^{2}d_{4}^{2}d_{2}w^{2}, d_{3}^{2}d_{4}^{2}d_{2}^{2}w, d_{3}^{2}d_{4}^{2}d_{2}^{2}w^{2}, d_{2}^{2}d_{3}^{3}d_{4}^{2}1

On the other hand the R-submodule V of M is much simpler, being generated by {1, d_{2}, d_{4}, d_{2}d_{4}}. Since every generator of U in the long list above is an R-multiple of a generator of M, we need only consider what R-multiples of the four generators 1, d_{2}, d_{4}, d_{2}d_{4} occur.

More explicitly, we are quoting the following:

Lemma A3

Let R be a ring and M a free R-module with basis {e_{1}, …, e_{n}}. Let V be the submodule generated by {e_{1}, …, e_{k}} for some k < n. Let U be an R-module generated by {r_{ij} · e_{i}}. Then

U ∩ V = J_{1} ⊕ ⋯ ⊕ J_{k} ⊕ 0 ⊕ ⋯ ⊕ 0,

where J_{t} is the ideal of R generated by {r_{tj}} (for fixed t, but j varying).

In any case, a brief examination of the list shows that the relevant multiples, or ideals as in the lemma are (1), (d_{3}), (d_{3}), (d_{3}). Thus, the R-submodule U ∩ V of M is generated by 1, d_{3}d_{2}, d_{3}d_{4}, d_{3}d_{2}d_{4}. Now all the R-modules involved are actually rings, and a set of R-module generators is also a set of ring generators, so we have shown what we claimed at the outset, namely that:

k[w, d_{2}^{2}, d_{4}^{2}](1, d_{3}, d_{3}d_{4}, t(t + w)d_{3}d_{4}) ∩ k[d_{2}, d_{3}, d_{4}] = k[d_{2}^{2}, d_{3}, d_{4}^{2}](1, d_{2}d_{3}, d_{3}d_{4}, d_{2}d_{3}d_{4}).

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