Stability of cohomology via double-coset products and suborbit diagrams

Stephen D. Smith and Kristin L. Umland ^{∗} Department of Mathematics (m/c 249) University of Illinois at Chicago 851 S. Morgan Chicago Illinois 60607–7045 USA

       e-mail:smiths@math.uic.edu and umland@math.uic.edu

Final version of 2 October 1995 (to appear in J. Algebra ) anon-ftp to russet.math.uic.edu, get /pub/smiths/dblcoset.latex.tar.Z

Abstract

Stability of subgroup cohomology is typically defined by a condition on individual double cosets, though use of the natural multiplication among them can reduce the amount of direct checking needed. The study of geometries for many finite groups, notably the sporadic simple groups, has frequently included the determination of suborbit diagrams, from which the needed multiplication coefficients are immediately available. The technique is applied to simplify certain calculations in the recent literature on the cohomology of the sporadic groups.

Introduction

The analysis in this note arose from a study of recent results in “exotic” finite group cohomology—starting with the determination by Adem-Maginnis-Milgram [AMM91] of mod-2 cohomology for the sporadic finite simple group M_{12}, where we saw an opportunity to streamline certain computations. The general double-coset algebra methods reviewed below are already known, see for example Cline-Parshall-Scott [CPS75, Sec 6]; the specific application of suborbit diagrams here appears to be new.

We focus on a particular result [AMM91, Thm 3.1] of Adem-Maginnis-Milgram (of a type which also seems common in the subsequent literature) showing that the intersection of the restrictions to the Sylow group from certain natural subgroups in fact contains no more than the restriction from the whole group; it is established there by checking stability under all the individual double cosets. Our main observation for such a situation is that point-line geometries give the centralizer-algebra approach a very concrete combinatorial interpretation—in which the parameters from the suborbit diagram afford a straightforward way of computing multiplicative consequences. These can greatly reduce the number of double cosets to be checked.

∗Both authors partially supported by NSA grant 904-93-H-3039

The organization of the paper is as follows: In Section 1, we review the underlying permutation representation on cosets of a subgroup, and the corresponding centralizer algebra defined by double cosets; we then describe an equivalent of the usual cohomological stability by means of a natural algebra homomorphism into Z. In Section 2, we axiomatize the intersection setup of [AMM91] in terms of verifying stability under an algebra-generating subset; in particular, we consider a sufficient condition for replacing the double cosets of the Sylow group with the (typically fewer) double cosets of a larger subgroup. We then interpret this condition in a standard context of point-line geometries for groups, and their suborbit diagrams. In Section 3, we demonstrate the streamlined double-coset technology in several cases where the suborbit diagram has been previously computed in the literature.

It is a pleasure to acknowledge conversations with many experts in group cohomology, including Alejandro Adem, Jon Alperin, Dave Benson, Lenny Evens, John Maginnis, Jim Milgram, and Steve Siegel. We also thank the referee for a number of helpful recommendations.

1 The centralizer algebra and stability, revisited

Since we hope this work will be of interest to researchers in both group geometries and group cohomology, we will try to present details which might not be familiar to each of these possible audiences. This section reviews the standard fact—see for example Cline-Parshall-Scott [CPS75, Sec 6] or Brown [Bro82, Exer III.10.2])—that stable cohomology of a subgroup can be characterized in terms of the corresponding centralizer algebra. Readers familiar with cohomology may wish to skim rapidly ahead to the latter (geometric) part of Section 2.

A “1-dimensional character” of the centralizer algebra

Throughout, G will denote a finite group, and H a subgroup. In this subsection, we wish to emphasize a point not explicitly made in the references [CPS75] [Bro82], namely the subalgebra property of the stabilizer in Lemma 1.2 below. So we develop here a convenient variant of those treatments based on the language of ring homomorphisms—indeed for purposes of cohomology with Z-coefficients, we will prefer to speak more formally of Z-algebra homomorphisms (and then p-adically of Z_{p}-algebra homomorphisms).

We use M as an abbreviation for the permutation module Z[G/H]; namely the free abelian group with the cosets as basis, and left multiplication by elements of G, giving the module structure under the group algebra ZG. In the algebra viewpoint, it is natural to denote the G-invariants (usually M^{G}) by M_{1}, since it is the full subset on which ZG acts in its trivial representation “1”. We will use H to denote the corresponding centralizer algebra 1 End_{ZG}(M), where End refers to the Z-algebra of endormorphisms of M as abelian group, while the subscript ZG indicates the subalgebra commuting with the ZG-module action. In view of this commuting, the endomorphisms of M in H must preserve the subspace M_{1} on which ZG acts as 1. In particular, the restriction Res_{M_{1}}^{M} : H → End(M_{1}) must induce a homomorphism of Z-algebras.

Now H is not a group algebra and so has no natural notion of “trivial” H-module; so we develop instead some parallel notation for the action it does have on M_{1}. It is elementary that transitivity of G on G/H (i.e., just one orbit) means that the submodule M_{1} of invariants has Z-rank just one; it consists of the Z-multiples of the formal sum m_{1} of all the cosets. As a result, any endomorphism α of M_{1} must act by simply multiplying the generator m_{1} by some integer, say λ(α); and the specification of this “eigenvalue” λ : End(M_{1}) → Z is naturally a Z-algebra isomorphism. Combining this with the map from H in the previous paragraph, we can define:

Lemma 1.1 The composition λ_{1} : H \underrightarrow{Res} End(M_{1}) →^{λ} Z is a Z-algebra homomorphism.

We may regard λ_{1} as the “character” of H determined by the submodule M_{1} on which ZG acts with its trivial character. Now in any H-module N—say with action given by ρ : H → End(N))—we can define an analogue N_{λ_{1}} of a trivial submodule, namely the Hsubmodule N_{λ_{1}} on which H acts via the homomorphism λ_{1}; equivalently, this is just the intersection over all T ∈ H of the kernels of the maps ρ(T) − λ_{1}(T)Id_{N}. Then given n ∈ N, we may wish to test it for membership in N_{λ_{1}}, by considering individual T ∈ H. If we set

Stab_{H}(n) =^{def} {T ∈ H : ρ(T)(n) = λ_{1}(T) · n},

then we see it is immediate, since ρ and λ_{1} are Z-algebra homomorphisms, that:

Lemma 1.2 Stab_{H}(n) is a Z-subalgebra of H.

And clearly n ∈ N_{λ_{1}} if and only if Stab_{H}(n) is the full algebra H. In particular, it will suffice to check that Stab_{H}(n) contains a subset generating H as Z-algebra.

Double-coset eigenvalues and stability

The aim of this subsection is to exhibit the above eigenvalue viewpoint in the context of a quick review of the standard Cartan-Eilenberg stability theory. Again for emphasis we develop a minor variant of the standard material, juxtaposing certain elements separated in the treatment of Brown [Bro82, Sec III.10].

First we recall that H^{∗}(H) can be regarded as H-module, summarizing a standard argument (e.g. [CPS75, Sec 6]): Let Z denote the integers Z, regarded as trivial ZG-module (and by restriction also as trivial ZH-module). Since H consists of endomorphisms of M, their action determines a corresponding action of H on Ext_{G}^{∗}(M, Z). But M is just the induced ZG-module Z ⊗_{ZH} ZG, so Shapiro’s lemma [Bro82, III.6.2] gives an isomorphism of Ext_{G}^{∗}(M, Z) with Ext_{H}^{∗}(Z, Z), which is just the group cohomology H^{∗}(H). Now using this module action, we can as in the previous subsection consider the submodule H^{∗}(H)_{λ_{1}} on which H acts via λ_{1}.

1For reference on centralizer algebras, see e.g. Curtis-Reiner [CR81, 11.22].

Next we examine the particular values taken by λ_{1} on the endomorphisms in H determined by the double cosets. Given a double coset HgH ∈ H\G/H, the map sending the trivial coset H = 1 · H into the sum of the (single) cosets in HgH extends uniquely to an endomorphism of M commuting with ZG-action via:

T_{HgH} : aH ↦ a( ∑_{g′H∈(HgH)/H} g′H).

(Compare also the “direct” definition in [CPS75, Sec 6]). Typically we will just write T_{g} for this operator when the subgroup H is fixed. It is also standard that the T_{g} (varying over all double cosets) afford a basis for H as free abelian group. Now since the generator m_{1} of M_{1} is just the sum of the single cosets, the eigenvalue λ_{1}(T_{g}) is just the number of single cosets in that double coset:

Lemma 1.3 λ_{1}(T_{g}) = |(HgH)/H| = |H : (H ∩ gHg^{−1})|.

We are also interested in the sum T_{G} of the operators T_{g} over all the double cosets. Since the sum of the values in Lemma 1.3 above just counts all the cosets, we see that λ_{1}(T_{G}) = |G : H|. Furthermore if the individual operators act with those values, then so does their sum, so we get an obvious inclusion:

H^{∗}(H)_{λ_{1}} ⊆ H^{∗}(H)_{λ_{1}(T_{G})} ,

where the latter is just the space where the single operator T_{G} acts as multiplication by |G : H|.

We can now summarize the relation of the centralizer algebra with stability by inserting the above inclusion into the middle of the standard Cartan-Eilenberg chain of containments:

Res(H^{∗}(G)) ⊆ H^{∗}(H)^{G/H−stable} ⊆ H^{∗}(H)_{λ_{1}} ⊆ H^{∗}(H)_{λ_{1}(T_{G})} ⊆ \frac{1}{|G : H|}Res(H^{∗}(G)).

In more detail: See e.g. [Bro82, Sec III.10] for the restriction homomorphism Res_{H}^{G} : H^{∗}(G) → H^{∗}(H), and the definition of stability of an element z ∈ H^{∗}(H). The inclusions among the first, second, and fifth terms above are the standard Cartan-Eilenberg relations—as seen just before, and in the proof of, [Bro82, III.10.3]. The inclusion of the second in the third is essentially the content of [Bro82, Exer III.10.2(a)(b)]; and the third in the fourth was noted in the previous paragraph. For the inclusion of the fourth in the fifth, it only remains to observe that it is the property of eigenvector for λ_{1}(T_{G}) which is actually used in the proof [Bro82, III.10.3] of the inclusion of the second in the fifth.

Of course in the usual situation of application, we are interested in some p and the ppart H^{∗}(G)_{p} of cohomology; so we choose H of coprime index p |̸ |G : H|, and the above inclusions all become equalities of Z_{p}-modules. Consequently from the viewpoint of the centralizer algebra, the usual definition of stability involving all the double cosets can be replaced by:

Lemma 1.4 If z is p-torsion and p |̸ |G : H|, then

z is stable  ⟺  T_{G}(z) = |G : H| · z = λ_{1}(T_{G}) · z.

Sometimes also it is useful to define for individual g ∈ G the notion of stable under g via the double coset HgH. In our language of eigenvalues, we get an analogous operatorwise characterization of stability:

Lemma 1.5 If z is p-torsion and p |̸ |G : H|, then

z is stable  ⟺  z ∈ H^{∗}(H)_{λ_{1}}.

An advantage of this latter formulation, in view of Lemma 1.2, is that it suffices to verify stability on an algebra-generating subset:

Lemma 1.6 For z ∈ H^{∗}(H),

(i) Stab_{H}(z) is a Z-subalgebra of H. In particular, if T_{g} acts via λ_{1}(T_{g}) on z, then so do all polynomials in T_{g}.

(ii) If z is p-torsion and p |̸ |G : H|, then for stability it suffices that:

Stab_{H}(z) _{Z}^{⨂} Z_{p} = H _{Z}^{⨂} Z_{p}.

Remarks: Readers familiar with representation theory may be accustomed to defining H with coefficients in the rationals Q rather than Z. But finite-group cohomology is all torsion in positive dimension, and so would be invisible with rational coefficients. For example, the Z-subalgebra Stab_{H}(z) is always likely to be “large”, even cofinite in H; but the measure of that finite index in suitable prime characteristics is ordinarily the crux of the stability problem.

We could also define a group stabilizer Stab_{G}(z) as those elements g ∈ G with T_{g} ∈ Stab_{H}(z). But since we have action by H and not by the group algebra ZG, this will only be a subset—not usually a subgroup—of G. We can at least say this set is a union of double cosets; which must be well known (Steve Siegel first showed us a proof). Of course we could further note that Stab_{G}(z) must generate some Z-subalgebra of Stab_{H}(z); but this might well be proper (indeed cofinite as above). In practice we typically would just check a few such T_{g} and hope to show that their algebra span really does cover all of H.

2 Pairs of subgroups and point-line geometries

From here on we always assume that p |̸ |G : H|, and consider algebras over the p-adic integers Z_{p}; as we are then interested only in the p-part H^{∗}(G)_{p}, typically we will for brevity omit the subscript p. In view of 1.5, we will always view the stable cohomology H^{∗}(H)^{G/H−stable} as H^{∗}(H)_{λ_{1}}.

The setup of [AMM91, Thm 3.1] is a particular case of a general situation of frequent interest: ^{2} Suppose we are given subgroups A and B intermediate between H and G; we may then wish to check whether stability under A and B implies stability under all of G. We will show how the sufficient condition emerging from the algebra viewpoint becomes particularly natural when A and B are the subgroups stabilizing a point and a line in a finite geometry for G.

First we fit such an intermediate pair A, B into our earlier language of eigenvalues. From H ⊆ A ⊆ G, we get a natural containment of coset spaces A/H ⊆ G/H and of double-coset spaces H\A/H ⊆ H\G/H; leading to corresponding inclusions of permutation modules M_{A} ⊆ M and centralizer algebras H_{A} ⊆ H. In particular, H_{A} is generated by the operators T_{a} with a ∈ A, and the subspace stable under A is given by H^{∗}(H)_{λ_{1A}}, where λ_{1A} denotes the restriction of λ_{1} to H_{A}. And of course similar remarks hold for B, with corresponding restriction λ_{1B} for H_{B}. The stable subspaces are clearly related by

H^{∗}(H)_{λ_{1}} ⊆ H^{∗}(H)_{λ_{1A}} ∩ H^{∗}(H)_{λ_{1B}};

and our subalgebra principle 1.6(i) gives a sufficient condition for the reverse inclusion:

Lemma 2.1 If the T_{a} and T_{b} generate H as Z-algebra, then H^{∗}(H)_{λ_{1}} = H^{∗}(H)_{λ_{1A}} ∩ H^{∗}(H)_{λ_{1B}}.

We remark that it is straightforward to develop analogues for the intersection of more than two subgroups. Indeed this is the case for the standard application of the approach in Cline-Parshall-Scott [CPS75, Cor 6.1] (a result independently obtained by Glauberman): Take G of Lie type in characteristic p with H a Borel subgroup B. There the double cosets are indexed by elements w ∈ W of the Weyl group, expressed as words in certain distinguished generators s ∈ W. The standard BN-pair axioms guarantee that T_{s}T_{w} = T_{sw} when sw > w in the length-ordering on W. From this it follows inductively that the T_{s} generate H as Z-algebra. And then 2.1 shows that cohomology of a Borel subgroup B stable under the minimal parabolics A_{s} = B ∪ BsB s must in fact be stable in the full group G.

This straightforward Lie-type calculation can be regarded as a model case: a principal motivation for the methods of this paper is to develop effective analogous computations for more general groups and geometries. In particular, the graph on the above double cosets with edges of form BwB − BswB is just the standard diagram for the length-ordering on W. The suborbit diagrams below can be regarded as partial analogues.

Double cosets of an intermediate subgroup

The proof of [AMM91, Thm 3.1] appears to work implicitly with double cosets not of H, but of the larger subgroup A. In the algebra viewpoint, it seems useful to develop this approach more explicitly. For the permutation module M on the cosets of H, we already defined notation H, λ_{1}, T_{HgH}; now we define corresponding notation for the module N on the cosets of A, “increasing each letter by 1” to get

2Recently Maginnis [Mag95] has developed another very natural technique, using control of fusion, which can often be applied in this situation.

I  the centralizer algebra of N
μ_{1}  the “character” of I on trivial submodule N_{1} = Z · n_{1}
U_{AgA} the generator of I corresponding to double coset AgA

And we explore how to modify 2.1 so that it can be sufficient to work just with the U_{g} ∈ I. We begin by developing a sufficient condition in terms of μ_{1}. The relation between the permutation modules M = Z[G/H] and N = Z[G/A] can be conveniently expressed by means of the sum T_{A} of the operators T_{a}, for we see that on the trivial coset H = 1 · H in M we have

T_{A} : H ↦ ∑_{a′H∈A/H} a′H.

The map identifying this latter sum with the coset A in N defines under G-translation an isomorphism of the image T_{A}(M) (a G-submodule since G commutes with H) with N. We use a tilde to denote the map T̃_{A} : M → N obtained by composing T_{A} with this isomorphism. In cohomology this determines the usual corestriction H^{∗}(H) → H^{∗}(A), and restriction is determined by the reverse map N → M given by

A ↦ ∑_{a′H∈A/H} a′H.

Thus the usual corestriction-then-restriction just corresponds to a breaking our original T_{A} into a composition of T̃_{A} followed by splitting A back into A/H. Note that by our hypothesis p |̸ |G : H| we also have p |̸ |A : H|; since T_{A} acts on the A-stable elements via |A : H|, we see that on them the corestriction T̃_{A} has a left inverse given by \frac{1}{|A:H|} times the restriction. The usual statement about stability under composition of restrictions now becomes in our language of algebra homomorphisms:

Lemma 2.2 If z ∈ H^{∗}(H)_{λ_{1A}} and T̃_{A}(z) ∈ H^{∗}(A)_{μ_{1}}, then z ∈ H^{∗}(H)_{λ_{1}}.

Proof: We see from the definition of the sum operators T_{G} and U_{G} that T̃_{A} intertwines them via:

T̃_{A}[T_{G}(−)] = |A : H| · U_{G}[T̃_{A}(−)],

so our hypothesis on z gives

T̃_{A}[T_{G}(z)] = |A : H|U_{G}[T̃_{A}(z)] =^{(1.4)} |A : H||G : A| · T̃_{A}(z) = |G : H| · T̃_{A}(z).

From the invertibility of T̃_{A} on H^{∗}(H)_{λ_{1A}} noted above, we get T_{G}(z) = λ_{1}(T_{G}) · z; hence again using 1.4, z is stable. □

In the setup of Lemma 2.1, we are in fact given z stable not just under the T_{a} as in Lemma 2.2, but also under the T_{b} for b ∈ B. So it is reasonable to hope that T̃_{A}(z) might also be stable under corresponding operators U_{b} for double cosets AbA of suitable b ∈ B; and even that these would generate I—giving G/A-stability of T̃_{A}(z) by Lemma 1.5, and then G/Hstability of z using Lemma 2.2. So next we will develop a sufficient condition corresponding to what is probably the most frequent situation in applications; namely when we have the intuitive formula “U_{AbA} = T_{A}T_{b}T_{A}”. Note that the formula is by no means a tautology—for example in case b = 1, the operator U_{1} is the identity on N, but T_{A}T_{1}T_{A} acts as multiplication by |A : H|^{2} on the isomorphic submodule T_{A}(M). More formally, this condition on some b is that the corestriction T̃_{A} should intertwine the maps U_{b} and T_{A}(T_{b}(−))—the latter considered on T_{A}(M). The formulation below may look rather technical; but in the next subsection we will see that it has a very straightforward combinatorial interpretation in the situation of a point-line geometry.

Lemma 2.3 Assume we are given some b ∈ B satisfying on T_{A}(M):

U_{b}[T̃_{A}(−)] = T̃_{A}[T_{A}(T_{b}(−))].

Then for z ∈ H^{∗}(H)_{λ_{1A}} ∩ H^{∗}(A)_{λ_{1B}}, we get U_{b} ∈ Stab_{I}(T̃_{A}(z)).

Proof: Viewing restriction and corestriction as in the previous paragraph, the hypothesis says in essence that the action of T_{A}(T_{b}(−)) on the submodule T_{A}(M) agrees with the action of U_{b} on N, identified with T̃_{A}(M). Now eigenvalues of operators are determined by subgroup indices as in 1.3, and we recall that these values are originally visible on the invariants M_{1} and N_{1} of the permutation modules. In particular we note that M_{1} lies in T_{A}(M), so that T_{A}(m_{1}) = |A : H| · m_{1}, and hence

T̃_{A}(m_{1}) = |A : H| · n_{1}.

So using stability of n_{1} and our intertwining hypothesis, we get

μ_{1}(U_{b}) · |A : H| · n_{1} =^{(1.5)} U_{b}[|A : H| · n_{1}] =^{(prev.line)} U_{b}[T̃_{A}(m_{1})] =^{(Hyp)} T̃_{A}[T_{A}(T_{b}(m_{1}))]

=^{(1.5)} T̃_{A}[λ_{1}(T_{A})λ_{1}(T_{b}) · m_{1}] = λ_{1}(T_{A})λ_{1}(T_{b}) · T̃_{A}(m_{1}) =^{(prev.line)} λ_{1}(T_{A})λ_{1}(T_{b}) · |A : H|n_{1}.

Of course the permutation module N is Z-free on the cosets, so we deduce from this calculation that the essential numerical content of our hypothesis on U_{b} is actually given by:

μ_{1}(U_{b}) = λ_{1}(T_{A})λ_{1}(T_{b}).

This says in fact that |A : A ∩ bAb^{−1}| = |A : H ∩ bHb^{−1}|, which is non-obvious (as noted before it fails if b = 1 and H ≠ A). Consequently when we apply the hypothesis to an element z taken from the A, B-stable cohomology we get:

U_{b}[T̃_{A}(z)] =^{(Hyp)} T̃_{A}[T_{A}T_{b}(z)] =^{(1.5)} T̃_{A}[λ_{1}(T_{A})λ_{1}(T_{b}) · z] =^{(prev.eqn.)} μ_{1}(U_{b}) · T̃_{A}(z),

which on using 1.5 with respect to G/A establishes the desired assertion. □

Remarks: Of course, whenever in applications we obtain as above some U_{b} in the stabilizer of z, by the observation 1.6(i) we know immediately that all powers of U_{b} also stabilize the corestriction of z. Typically this gives a shortcut via A over Lemma 2.1 using H.

The indicated sufficient condition on U_{b} is far from necessary, and is only chosen as the most likely application. From the proof it should be clear that it is only necessary for T̃_{A} to intertwine U_{b} with some element in the algebra-span of the T_{a} and T_{b}.

Point-line geometries and suborbit diagrams

The viewpoint of finite geometry has been increasingly exploited in work on group cohomology; see for example Quillen’s discussion of Tits buildings in [Qui78, Sec 3] and Webb’s alternating-sum expression [Web87] over subgroup complexes more general than those of Brown and Quillen. For Lie-type groups, the relevance of the building geometry in particular to the centralizer algebra is explicitly discussed in [CPS75, Sec 6]. For sporadic groups, the analogous finite geometries also feature prominently in [AMM91] and subsequent work. In this subsection, we give more concrete connections of fairly general geometries with the centralizer algebra viewpoint. The methods can illuminate many otherwise technical numerical and group-theoretic statements (and in particular can replace or at least reduce machine computations).

First we will show how certain commonly-occurring geometric features lead to an operator U_{b} automatically satisfying the stability condition 2.3. We begin by reviewing some standard features of modern work on group geometries. This material may be less familiar to cohomologists, but now group theorists may wish to skip ahead.

A good background source for the main context of coset geometries is the influential work of Tits [Tit81]. We regard the cosets of our previous intermediate subgroups A and B as the vertices of a bipartite graph, joined by an edge whenever two cosets intersect in a coset of A ∩ B; henceforth we assume our H not only contains, but actually equals this intersection. Our full group G then acts edge-transitively on this graph.

By this transitivity, all B-cosets in the graph are adjacent to the same number of Acosets. Often that common valency is of form q + 1 for a prime power q; and then we usually think of the cosets of A and B as projective points and lines over the finite field F_{q}. We can think of a line as being defined by the subset of its points; and the graph edges can be regarded as projective point-line flags. (It may or may not be the case that our whole graph actually does arise from some higher-dimensional projective space). To obtain more geometric notation under this assumption, we will now identify the trivial A-coset A with a particular point P, and regard the group A as its stabilizer G_{P}; and similarly the trivial coset B will determine a particular line L, stabilized by the group B = G_{L}. Then H = A∩B is the stabilizer G_{P L} of the flag P L. Most often such point-line geometries are distinguished by the further features:

Hypothesis 2.4

(a) The line stabilizer G_{L} acts doubly-transitively on the points of L.

(b) Two collinear points lie on just one line.

Condition (a) will hold in the frequent case that G_{L} induces the action of P SL_{2}(q) on the q + 1 points of the projective line over F_{q}—indeed this is usually the real motivation for the terminology of points and lines (as opposed to more general block designs). It occurs for example in all untwisted Chevalley groups, and in many geometries for sporadic groups (including the examples in the next section). The condition (b) is similarly common, and is particularly well-known as (LL) in Tits [Tit81].

We show that this standard geometric situation guarantees that the operators U_{b} are stabilizing in the sense of the previous subsection:

Lemma 2.5 Hypothesis 2.4 implies Lemma 2.3 for U_{b} with any b ∈ B\H.

Proof: We will verify the hypothesis of Lemma 2.3, first translating the action of the relevant operators into our geometric notation.

The trivial H-coset H is geometrically the point-line flag P L (stabilized by the subgroup H = G_{P L} = A ∩ B). The action of A on the cosets A/H becomes the transitive action of the point stabilizer G_{P} on all the lines L′ containing P. So the sum-operator T_{A} has the geometric expression:

T_{A}(P L) = ∑_{L′∋P} P L′.

Since T_{A} commutes with the G-action, we get analogous statements for the other points. Recall next that we wrote the operator T_{A} on M as the corestriction T̃_{A} into N, followed by restriction splitting A back into its H-cosets. In geometric notation the corestriction has the simple expression:

T̃_{A}(P L) = P ,

corresponding to our convention that the point P denotes the trivial A-coset A in the permutation module N; while the restriction sends P back to the above sum for T_{A}(P L). In particular the image of T̃_{A} is independent of the line of the flag; it will be useful later to note that:

T̃_{A}(T_{A}(−)) = |A : H|T̃_{A}(−).

To describe T_{b}, we apply the hypothesis (a) of double transitivity: this means that there are just two double cosets of H in B—namely H, and HbH (for any b ∈ B\H) containing all the other single H-cosets. Geometrically these correspond to the flag P L, and the flags P′L for all the other points P′ on L. So the non-trivial double-coset operator has the action

T_{b}(P L) = ∑_{P ≠P′∈L} P′L.

Again T_{b} commutes with the G-action so we get a similar statement for any other line.

We now have the ingredients to compute the right side of the hypothesis of Lemma 2.3. We get a generator of T_{A}(M) by applying T_{A} to P L, namely

∑_{L′∋P} P L′.

Then applying T_{b} to each line on P sums over the other points of those lines, giving

∑_{L′∋P} ∑_{P′≠P ∈L′} P′L′.

Finally we apply the composition T̃_{A}T_{A} as above (with the earlier simple expression for T̃_{A} to get:

|A : H| ∑_{L′∋P} ∑_{P′≠P ∈L′} P′.

At this point, we observe using hypothesis (b) that the point P′ uniquely determines the line L′ on P; consequently each point collinear with P occurs just once inside the above sum.

To compute the right side of the hypothesis of Lemma 2.3, using again our remark on the composition T̃_{A}T_{A} we have |A : H|U_{b}(P). Now by definition U_{b} takes the trivial A-coset A, corresponding geometrically to P, into the sum of all the single cosets in the double coset AbA. Since this double coset contains the cosets in HbA, corresponding to the other points of the line L, we see by edge-transitivity that the sum must consist exactly of those cosets for the points collinear with P. So this does agree with our computation of the previous paragraph, and the proof is complete. □

From the proof it should be clear that we could actually weaken the hypothesis (b) to the condition that the (fixed) number of lines on a pair of collinear points is coprime to p. This occurs for example in the geometry of Neumaier [Neu84] for the alternating group G = A_{7} and p = 2, where two collinear points determine 5 lines.

In the remainder of the subsection, we indicate how the multiplicative consequences of the above U_{b} can be computed using the parameters of the suborbit diagram.

We begin with a review of the conventions for such diagrams. It is elementary that the double cosets of A correspond with the suborbits, namely the orbits of the point stabilizer G_{P} = A on the set of all points. For example, under our hypothesis 2.4(a) the double coset AbA corresponds to the suborbit of all points P′ collinear with P. We can define an associated collinearity graph on the points alone, where edges are given by the collinearity relation.

The corresponding suborbit diagram encodes via its parameters the relationships in the quotient of this graph by the action of A. In overall summary: We draw a circle for each suborbit, and inside write its size. We connect two circles with an edge whenever there are collinearities between their members. At the junction of such an edge with a suborbit (say for AxA) we count the number of points in the suborbit (say for AyA) at the other end of the edge which are collinear with a fixed point from the original AxA. By transitivity of A on the suborbits, this value does not depend on on the choice of fixed point.

Now we will illustrate that construction in further detail. We begin with the trivial suborbit for P itself; thus for double coset A = A1A we draw a circle with the size 1 inside. This point P is collinear only with the points of the suborbit corresponding to AbA. It will be convenient to develop some further notation for sizes. Let q+1 denote the number |B : H| of points on a line, and r + 1 the number |A : H| of lines on a point. Then (using hypothesis 2.4(b) as before for uniqueness) the number of points collinear to P is (r + 1)q. So this is the size we write below (abbreviated for convenience as C) in the circle for AbA; it is also the number of connections from A to AbA, so we write it at that end of the corresponding edge. We turn our attention to edges emanating from the circle for AbA. A point P′ is certainly connected to the unique point P of the previous suborbit, so we write a 1 at this end of that edge. Next observe that the line determined by P, P′ (in view of 2.4(b) uniquely) has (q + 1) − 2 = q − 1 further points, all in this suborbit AbA. Below we will not draw a loop representing self connections; instead we will just write the corresponding number above the circle. Thus at the beginning of the suborbit diagram we have the structure:

[figure: 1; C; A; C; 1; (q − 1) + …; n; AbA; …]

Here for convenience we have drawn just one further edge out from AbA, though in general there might be connections to more suborbits. For each such edge, the number n above would give for P′ the number of collinear points in that adjacent suborbit. And of course such lines might intersect AbA in further points, which would add to the value q − 1 written above the circle. One proceeds in this way until all edges and corresponding values are filled in. The process is a straightforward algorithm based on knowledge of the geometry, and has been carried out in the literature for many interesting permutation geometries (and permutation representations G/A).

We now indicate how these paramenters allow us to compute the algebra product of U_{b} with the operator U_{d} for some suborbit AdA. Suppose that the parameters for the suborbits adjacent to AdA are denoted by:

[figure: …; …; s; AcA; …; …; t; …; AdA; …; …; u; …; AeA]

Here again for convenience we show only two adjacencies, though in general there could be more. With the above notation, it is standard that the composition of the collinearity operator U_{b} with U_{d} is given by the parameters at the far end of edges:

U_{b}U_{d} = sU_{c} + tU_{d} + uU_{e}.

Now if we have as earlier shown that U_{b} ∈ Stab_{I}(T̃_{A}(z)) then by the subalgebra property 1.6 we know also that (U_{b})^{2} also lies in the stabilizer; so with d = b and c = 1 we see from the right side of the formula (since U_{1} and U_{b} are already in the stabilizer) that if u is invertible in Z_{p} then also U_{e} lies in the stabilizer. We could continue, applying U_{b} to U_{e} and examining the right side; ideally we might even force all operators in turn into the stabilizer.

3 Sample applications

Now we are ready to demonstrate the methods in practice. We consider several stability results already in the literature on cohomology of sporadic groups, comparing our algebra techniques with the original calculations. The method is also applicable for many sporadic groups whose cohomology is still not known, and indeed is being applied in work still in progress.

As in the previous section, we assume a particular prime p has been fixed, and work only with the p-part of cohomology; we continue omit the subscript p, which should be regarded as understood throughout the section.

M_{22} for p = 2

The simplest non-trivial case of the above method occurs when there is just one further double coset beyond A and AbA, namely when G acts with permutation-rank 3 on the cosets of A.

We take G to be the sporadic Mathieu group M_{22}, and consider a very specific piece of the work in the calculation by Adem-Milgram [AM95] of its mod-2 cohomology. For A, B we take the subgroups they denote by G_{1}, G_{2}, with structure 2^{4}A_{6} and 2^{4}S_{5}; and then H is the intersection A ∩ B of structure 2^{4}S_{4}. In fact Adem and Milgram do not explicitly state an analogue of [AMM91, Thm 3.1]. However, they do determine the restrictions from A and B in [AM95, 2.9,2.10] and then use them in going on to the stable cohomology of a Sylow 2-group inside H. Furthermore they describe in [AM95, 1.2] the underlying calculation on the double cosets of A (as being obtained by C. Overton on a computer). So this seems a natural situation in which to demonstrate our geometry/algebra methods by establishing the implicit intersection-result in the form 2.1. The actual double-coset calculation will be a one-line proof.

We must first establish the geometric Hypothesis 2.4.

To describe the geometry, we extract some features from Ronan-Smith [RS80]. As is often the case, the subgroups A, B, H are self-normalizing; so it is standard that the permutation representations on cosets are equivalent to the representations on conjugates of these subgroups. And here the conjugates correspond with certain natural structures in the permutation representation of G on 22 “letters”, giving a particularly concrete expression for the geometry. First, a point can be regarded as a hexad—namely one of the special 6-subsets of 22 defining the Steiner system S(3, 6, 22) (see for example Conway [Con71, p.225]). And a line can be regarded as a quintet, namely a suitable partition of the 22 letters with parts of size 2, 4, 4, 4, 4, 4. A point is then on a line when the 6-set is the union of the 2-set with one of the 4-sets in the partition.

The index |B : H| = |S_{5} : S_{4}| = 5 records the obvious geometric statement that a line as above has 5 points; and the group S_{5} in the line stabilizer B in fact acts in its natural permutation representation on these 5 points—of course this action is (more than) doubly transitive, giving Hypothesis 2.4(a). (For consistency with our earlier remark, we can also regard S_{5} as P GL_{2}(4) acting on the 5 points of the projective line over F_{4}). Hypothesis 2.4(b) also holds, since two different lines on a pair of points would lead to two different hexads intersecting in more than 3 points, contradicting the definition of Steiner system S(3, 6, 22). (Viewed another way, the situation would produce a vector of weight below the known minimum for the subspace of the binary Golay code on which M_{22} acts).

Consequently for b ∈ B\H we know by Lemma 2.5 that U_{b} stabilizes the intersection of restrictions from A and B. We remark similarly that |A : H| = |S_{5} : A_{4}| = 15 indicates that a point lies on 15 lines. In our further notation for suborbit diagram parameters in the previous section, we have q = 5 − 1 = 4 and r = 15 − 1 = 14; and there are then C = 60 = 15 · 4 points collinear to fixed P, giving the double coset AbA corresponding to the operator U_{b}. It is well known that M_{22} acts with permutation rank 3 on the 77 points; in particular, A is also transitive on the remaining 16 = 77 − 60 − 1 points not collinear with P, so there is only one more suborbit AcA for some c. Indeed even the suborbit diagram is well known in the rank-3 literature; in [Bue83, Ex.3] Buekenhout gives the Levi diagram of suborbits on point-line flags, from which the point-suborbit diagram is immediate:

[figure: 1; 60; A; 60; 1; 47; 12; AbA; 16; 45; 15; AcA]

As noted in the previous section, these numbers translate into multiplication coefficients via:

(U_{b})^{2} = 60U_{1} + 47U_{b} + 45U_{c}.

This one line essentially proves the desired result; but perhaps in this first application we should review the details. We got U_{b} in the stabilizer of the intersection from the geometry, and the subalgebra property 1.6 puts the left side in the stabilizer. On the right side, we already have the identity U_{1} and also U_{b} in the stabilizer, so also 45U_{c} is in the stabilizer. As we are working 2-adically and 45 is odd, we must have U_{c} itself in the stabilizer. Now all three operators spanning I are in the stabilizer. So using Lemma 2.2 we conclude the intersection is indeed stable.

M_{12} for p = 3

The 3-part of the cohomology of the sporadic Mathieu group M_{12} is determined by Milgram and Tezuka in [MT95]. As in the previous example, an intersection-result is relevant to part of that work; we show here that most of the work of establishing Lemma 2.1 can be straightforwardly checked using the suborbit diagram.

Much as in the previous example, the 3-local geometry [RS84, p 82] has a concrete description in terms of subgroups stabilizing partitions of the 12 letters of the standard permutation representation of M_{12}; so we start with that viewpoint, to lead up to suitable sugbroups A, B, H. We consider partitions of twelve consisting of a 3-set and a 9-set (denoted (3|9)) to be points, and partitions consisting of four 3-sets (denoted (3|3|3|3)) to be lines (with the added stipulation that any pair of 3-sets in a (3|3|3|3) partition form a special 6-set or “hexad” in the Steiner system S(5, 6, 12)) [Con71, p 225]. We define incidence between points and lines in an obvious way, i.e. a point (3|9) is on a line (3|3|3|3) if and only if the 3-set associated with it is one of the 3-sets of the line. From this it is clear that in this geometry there are exactly four points on every line, and with a calculation in the Steiner system one checks there are exactly four lines on every point. (Thus q = r = 3 in our earlier notation).

The group M_{12} is of 3-rank 2, with 3-Sylow subgroup an extraspecial group of order 27 and exponent 3. If we consider a fixed point say P = (1 2 3 |4 5 6 7 8 9 10 11 12), then the stabilizer G_{P} of P in M_{12} has structure 3^{2}GL_{2}(3). It turns out that a line stabilizer, G_{L}, where L contains P (say L = (1 2 3 |4 5 6 |7 8 9 |10 11 12)) is also a 3^{2}GL_{2}(3). Moreover, their intersection is the normalizer of a 3-Sylow subgroup. We will set G_{P L} = G_{P} ∩ G_{L}, and S equal to the 3-Sylow subgroup, of index 4 in G_{P L}.

In the notation of Section 2 we set G = M_{12}, A = G_{p}, B = G_{L}, and H = G_{pL}. We want to establish the intersection-result 2.1 via 2.2; again we start by showing U_{b} is in the intersection stabilizer because our geometry satisfies the sufficient conditions of Lemma 2.5. Part (a) of the corresponding Hypothesis 2.4 follows from the fact that G_{L} = 3^{2}GL_{2}(3), where on the points of L, GL_{2}(3) has its usual 2-transitive action on the 4 points of the projective line over F_{3}. To see why part (b) of Hypothesis 2.4 is true, consider two points P_{1} and P_{2} on two lines. Then two of the 3-sets of each line would be the 3-sets of the points P_{1} and P_{2}. Some pair Q, R of the remaining 3-sets, one from each line, would have to intersect in at least two letters—and thus in all three, since by definition of S(5, 6, 12) a 5-set determines a unique hexad, and the hexads P_{1} ∪ Q and P_{1} ∪ R already share 5. Now Q = R means that the two lines were identical. Thus (b) follows. Consequently by 2.5 we have that U_{b}, the operator associated with the only non-trivial double coset in H \ B/H, lies in Stab_{I} T̃_{A}(z) for any z in the intersection I of Lemma 2.1; or more generally, U_{b} is in Stab_{I}(I).

We get a point-suborbit diagram for A = G_{p}, again deduced from the published point-line diagram [Bue82, Sec 9]:

[figure: A; 1; 12; AbA; 12; 2; 1; 9; AcA; 108; 2+4; 1; 4; 1; AdA; 27; .; 4; 8; AeA; 72; 3; 6; 3]

As noted earlier, this allows us to compute the product of U_{b} with the other operators:

(i) U_{b}U_{b} = 12U_{1} + 2U_{b} + U_{c} (ii) U_{b}U_{c} = 9U_{b} + 6U_{c} + 4U_{d} + 6U_{e} (iii) U_{b}U_{d} = U_{c} + 3U_{e} (iv) U_{b}U_{e} = 4U_{c} + 8U_{d} + 3U_{e} Now since U_{1}, U_{b} and thus (U_{b})^{2} lie in the subalgebra (call it S) stabilizing I, we get that also U_{c} ∈ S from (i) above. Using this in (ii), and removing terms for U_{b}, U_{c} already known in S, we next get that 4U_{d} + 6U_{e} ∈ S. We apply U_{b} to this combination using (iii),(iv); on removal of U_{b}, U_{c} terms we get 48U_{d} + 30U_{e} ∈ S. Subtracting 5(4U_{d} + 6U_{e}) leaves 28U_{d} ∈ S. Since we are interested only in the 3-part of cohomology, coefficients coprime to three are invertible in the centralizer algebra over Z_{3}, so in fact we have U_{d} ∈ S. However, U_{e} never occurs in products with a coefficient that is prime to three; so the subalgebra generated by the other 4 operators contains 3U_{e} but not U_{e}, and algebra methods will be of no further help.

To finish, we work with the original notion of stability under e. The concrete view of the geometry will allow us to do this directly, avoiding extensive machine computation.

We want to pick a suitable representative e and point Q (the image of P under e) in AeA, and explicitly find the intersection A ∩ eAe^{−1} = G_{P} ∩ G_{Q}. We can think of the points in the various suborbits as having certain intersection properties with the 3-set of P. In particular, the 3-set of any point in AbA has empty intersection with the 3-set in P, precisely because they are collinear. Recall that this pair of 3-sets forms a hexad of the Steiner system. On the other hand, the points in AeA are those whose 3-sets have empty intersection with the 3-set of Q, but the pair does not form a hexad in the Steiner system. Since we can choose a particular Steiner system S(5, 6, 12), we will take (1 2 3 4 5 6) to be a hexad; and since a 5-set determines a unique hexad (in this case (1 2 3 4 5)), (1 2 3 4 5 7) is not in our Steiner system. If Q = (4 5 7|...), then P and Q are not collinear. Thus an element of G_{P} ∩G_{Q} must stabilize (1 2 3|4 5 7|...). The following computations can be done by hand, but we found it easier (and more reliable) to work with M_{12} in the group theory language GAP [SBB^{+}92]. We check that the intersection G_{P} ∩ G_{Q} has structure S_{3}. Furthermore we will construct an involution y that interchanges P and Q, and lies in the centralizer of the 3-Sylow subgroup of G_{P} ∩G_{Q}; we will take this to be our double coset representative for AeA = AyA. We first design an involution x = (1, 4)(2, 5)(3, 7)(6, 8)(9, 10)(11, 12) to lie in M_{12} and interchange P and Q; but find that it instead inverts that Sylow 3-group = < (1, 2, 3)(4, 7, 5)(6, 12, 10)(8, 9, 11) >. However, w = (2, 3)(5, 7)(6, 12)(8, 11) is an involution in G_{P} ∩ G_{Q}, so also inverts that 3-group; then the involution y = xw = (1, 4)(2, 7)(3, 5)(6, 11)(9, 10)(8, 11) centralizes the 3-group and clearly takes P to Q.

A standard argument finishes the proof: In the usual definition, z ∈ H^{∗}(A) is stable ^{3} under y ∈ G if

Res_{A∩yAy^{−1}}^{A}(z) = Res_{y^{−1}Ay∩A}^{A}(yz),

where yz denotes the conjugation action of y on z. In our case y is an involution so y = y^{−1}; and it interchanges P and Q so that both the indicated intersections are G_{P} ∩ G_{Q}, which y then normalizes. But since the 3-part of cohomology is always determined by restriction to a Sylow 3-subgroup, and y centralizes a Sylow 3-subgroup of that intersection, y acts as the identity on the restriction of z to H^{∗}(G_{P} ∩ G_{Q}). We conclude in fact that y stabilizes all the cohomology of G_{P} = A, including the image of our intersection I under T̃_{A}. In particular the operator U_{e} = U_{y} will lie in the stabilizer S of T̃_{A}(I). As we earlier obtained the other operators, we now have that the whole centalizer algebra I stabilizes anything in T̃_{A}(I). In view of 2.2, the desired intersection-result is proved.

Finally, though we have proven that I = Im(Res_{G_{pL}}^{G}), we really want Im(Res_{S}^{G}) for the Sylow group S. But this follows from the standard stability characterization [Bro82, III.10.3] since we have the additional condition that S < G_{pL}. More explicitly, we have that

3invariant in Brown [Bro82, III.10].

I = Res_{G_{pL}}^{G_{p}}(H^{∗}(G_{p})_{(3)}) ∩ Res_{G_{pL}}^{G_{L}}(H^{∗}(G_{L})_{(3)}) = Res_{G_{pL}}^{G}(H^{∗}(M_{12})_{(3)}).

By the standard result

H^{∗}(G_{pL})_{(3)} ≅ H^{∗}(S)^{G_{pL}/S−stable} = Res_{S}^{G_{pL}}(H^{∗}(G_{pL}))_{(3)}

so by applying Res_{S}^{G_{pL}} to both sides of the above equation for I we get

Res_{S}^{G_{pL}}(I) = Res_{S}^{G_{pL}}(Res_{G_{pL}}^{M_{12}}(H^{∗}(M_{12})_{(3)}))
= Res_{S}^{M_{12}}(H^{∗}(M_{12})_{(3)})
≅ H^{∗}(M_{12})_{(3)}

by transitivity of the Res operator. Thus we have the corresponding result for the Sylow group S.

M_{12} for p = 2

As a final example it seems appropriate to consider the case of M_{12} with p = 2, specifically the result [AMM91, Thm 3.1] in Adem-Maginnis-Milgram which originally inspired these observations. It is only a little more complicated than the above examples, illustrating how the subalgebra method can be used repeatedly, to build up the known part of Stab_{I}(T̃_{A}(z)) in several steps until all of I is reached.

For A and B we take the subgroups denoted W and W′ in [AMM91]; our H = A ∩ B agrees with their notation H for a Sylow 2-subgroup. Here A has structure 2^{1+4}S_{3} and is the stabilizer of the point defined by the central involution of the Sylow group H; while B is of structure 2^{2+1+2}S_{3}, and is the normalizer of a line given by a suitable 4-group of such involutions. As in the previous subsection we get a concrete view of the geometry from the permutation represention on 12 letters: we can identify points with the 4-subsets of the 12 letters, and lines with certain 4, 4, 4-partitions. Again there is a connection with the Steiner system S(5, 6, 12): the involution defining a point fixes the corresponding set P of 4 letters; and then P, with each of the 2-cycles of that involution, defines a special hexad.

We see have 3 = |A : H| points per line, with S_{3} in A acting in its natural permutation representation doubly-transitively, giving Hypothesis 2.4(a). (Again, we can regard S_{3} as GL(2, 2) acting on the 3 points of the projective line over F_{2}). And Hypothesis 2.4(b) is more immediate: since a line is a Klein 4-group whose 3 involutions are the points, it is clear that the product of any two of the points is the third—and thus two collinear points determine the rest of the (unique) line on them. (Though again this could also be established with an argument using the Steiner system). So as an immediate consequence of Lemma 2.5, we know that the operator U_{b} lies in the subalgebra S stabilizing the intersection I of restrictions from A and B. This avoids one of the larger computer calculations in [AMM91] (where the double coset AbA is given by a representative called g_{10}).

We turn to the suborbit diagram to examine the multiplicative consequences of U_{b} ∈ S. In fact there are 11 double cosets of A, denoted in [AMM91] via explicit (randomly computergenerated) representatives denoted g_{0} = 1, g_{1}, …, g_{10}. For ease of comparison we will use their numbering; and further abbreviate each operator U_{g_{i}} by U_{i}. (So now the identity operator for the trivial double coset A will be denoted by U_{0} rather than U_{1} as before). The suborbit diagram is well known: again Buekenhout in [Bue82, Sec 7] gives the Levi diagram for point-line flags, from the the point-suborbit diagram follows. In fact that diagram is for the automorphism group M_{12} ·2 with only 8 suborbits; it is straightforward to split 3 of these into pairs (namely for g_{4}, g_{8} and g_{7}, g_{1} and g_{5}, g_{6}) exchanged by an outer autormorphism of M_{12}. In the diagram below, such an outer automorphism rotates around the horizontal axis:

[figure: U_{0}; 1; 6; 1; U_{10}; 6; 1; 4; 1; U_{2}; 24; 1; 2; 1; 2; 1; U_{4}; 48; 2; 1; 1; 1; 2; 3; U_{8}; 48; 2; 1; 2; 3; U_{7}; 32; 3; 1; U_{1}; 32; 3; 1; U_{6}; 96; 2; 2; 2; 2; U_{5}; 96; 2; 2; U_{3}; 96; 1; 1; 6; U_{9}; 16]

So we begin by determining the multiplicative consequences of U_{b} = U_{10} ∈ S. From

U_{10}U_{10} = 6U_{0} + U_{10} + U_{2}

we get U_{2} ∈ S. Next

U_{10}U_{2} = 4U_{10} + U_{2} + (U_{4} + U_{8})

gives U_{4} + U_{8} ∈ S. Then

U_{10}(U_{4} + U_{8}) = 4U_{2} + (U_{4} + U_{8}) + 3(U_{7} + U_{1}) + (U_{6} + U_{5})

from which we get 3(U_{7}+U_{1})+(U_{6}+U_{5}) ∈ S. Now further products lead only to combinations in the span of these, together with even multiples of generators; since we are interested in the 2-part of cohomology, we now have all the multiplicative consequences of U_{b} = U_{10}.

We check next that it suffices, again via multiplicative consequences, to verify directly that just the three further operators for g_{4}, g_{7}, g_{9} lie in S. For assuming U_{4} ∈ S, from the earlier (U_{4} + U_{8}) ∈ S. we get U_{8} ∈ S. (Or, we could use conjugacy of U_{8} and U_{4} under the outer automorphism of M_{12}). Also we can look just at the U_{4}-part of an earlier product:

U_{10}U_{4} = 2U_{2} + U_{8} + 3U_{7} + U_{6}

to get 3U_{7} + U_{6} ∈ S (and similarly 3U_{1} + U_{5} using the other half). If we now also assume U_{7} ∈ S, we get from the above 3U_{7} + U_{6} that U_{6} ∈ S. Action of the outer automorphism similarly gives U_{1}, U_{5} ∈ S. Finally assume we also have U_{9} ∈ S. From

U_{10}U_{9} = U_{3}

we get U_{3} ∈ S. As we have obtained all 11 of the basic operators, S is all of the centralizer algebra I, and we conclude that the intersection I is indeed G-stable.

We remark also that the subalgebra generated by the 9 operators other than U_{4}, U_{8} contains 2U_{4} and 2U_{8} and U_{4} + U_{8}, but not the individual operators U_{4}, U_{8}. So any minimal subset of operators to check must include one of U_{4}, U_{8}.

Of course stability for g_{4}, g_{7}, g_{9} (among the others) is established computationally in [AMM91]. However we will also indicate here how viewing the points concretely as involutions (or 4-sets) leads to a direct natural construction of the intersections required for the stability verifications. So with P the point fixed by A, let z_{P} be the involution of the center of A. Correspondingly let Q denote a point of Ag_{i}A, with involution z_{Q}, as we consider each of i = 4, 7, 9 in turn below. Then A ∩ g_{i}Ag_{i}^{−1} = G_{P} ∩ G_{Q}, which we will abbreviate by G_{P Q}. In each case we try to choose a representative g_{i} for Ag_{i}A which interchanges P and Q, so that conjugation by g_{i} in fact normalizes G_{P Q}.

First consider g_{9}. Here the product z_{P} z_{Q} has order 3 and lies in the conjugacy class of M_{12} denoted 3B in the Atlas [C^{+}85]. Then z_{P} , z_{Q} generate a dihedral subgroup D of order 6; let E denote its normal subgroup of order 3. From the Atlas, we have N_{G}(E) of structure S_{3} × A_{4}. If z_{P} , z_{Q} had non-trivial projection in the A_{4}, we would get C_{G}(D) of order 4; whereas this group is also G_{P Q}, which we know from the diagram has order 12. So z_{P} , z_{Q} generate the direct factor S_{3} above, and G_{P Q} is given by the factor A_{4}. Now the third involution z_{P} z_{Q}z_{P} of the S_{3} switches P and Q, and centralizes G_{P Q}. So we may re-choose g_{9} to be this third involution; its conjugation action on the intersection G_{P Q} is trivial, which shows that it stabilizes all of H^{∗}(A), including the image of the desired intersection I.

In case of g_{7}, the product z_{P} z_{Q} lies in the class 3A. Again let D denote the dihedral group of order 6, and E its subgroup of order 3. This time the Atlas shows N_{G}(E) has structure 3^{1+2} : (2×2) (the Sylow 3-normalizer of the previous example). In a Sylow 2-group (containing z_{Q}) in this normalizer, we may choose the unique involution f centralizing E. Then G_{P Q} is of structure S_{3}, generated by f and the unique subgroup of order 3 centralized by z_{Q} in the Sylow 3-group of the normalizer. Again the third involution z_{P} z_{Q}z_{P} of D interchanges P and Q; and it lies in D which centralizes G_{P Q}. So just as before, it stabilizes all of H^{∗}(A).

In case of g_{4}, the product z_{P} z_{Q} has order 4, so that these two involutions generate a dihedral group D of order 8; this time let E denote its center of order 2. The involution (z_{P} z_{Q})^{2} is z_{R} for a point R at distance 2 in the point-graph from each of P, Q. If F denotes the subgroup of order 2 generated by z_{R}, then the intersection G_{P Q} is given by E × F. The other involutions of this group correspond to points X, Y which with R define a line of the geometry; furthermore X, Y are collinear with P, Q respectively. Now it is not hard to directly find a new choice of g_{4} which interchanges P and Q. (We remark that Adem-Maginnis-Milgram [AMM91, p 107] indicate “we can replace”, and then implicitly do rechoose so as to replace their original G_{g_{4}} (in our notation G_{P Q}) by “conjugate number 10” in their Appendix; giving the subsequent identification of this new choice as ⟨a^{2}[z, zz], z⟩ in their notation).

Notice however that g_{4} interchanging z_{P} and z_{Q} must centralize (z_{p}z_{Q})^{2} = z_{R}; thus g_{4} fixes R, and hence must also interchange the involutions z_{X} and z_{Y}, since these determine the unique graph-paths from X, Y to R. So in contrast to the previous calculations, conjugation by g_{4} will not stabilize all of H^{∗}(A). The remaining argument in [AMM91] finally makes non-trivial use of the fact that we examine g_{4}-conjugation only on α ∈ H^{∗}(H) which is A, Bstable. (Here it is convenient to work at the level of H rather than A). Further geometric analysis will also help make that argument more explicit.

First, we may as well assume that the line on P and X is the original line L stabilized by B; this essentially corresponds to the choice of conjugate 10 above. We can then choose an involution b ∈ B = G_{L} interchanging P and X. Then b normalizes the intersection H ∩ bHb = G_{P L} ∩ G_{XL}, which we will denote by G_{P XL}. Notice now that G_{P Q} centralizes z_{P} and z_{X} ∈ E, and so lies inside G_{P X}; but G_{P X} = G_{P XL}, since we verified that 2 collinear points are on just one line. Thus our group G_{P Q} lies inside G_{P XL} which b normalizes; furthermore it is on this intersection G_{P XL} that b-action must fix the restriction of α, by the hypothesis of B-stability. Consequently if we conjugate our entire stability question by b, we see it will suffice to show that bg_{4}b-action fixes Res_{bG_{P Q}b}(α).

Now b has conjugated z_{X},z_{R} to z_{P},z_{b(R)}; so that P and b(R) must determine one of the other lines L′ on P. (This corresponds to “conjugate to a subgroup of K” in [AMM91, p.107]). Now we can choose an involution a ∈ A = G_{P} interchanging L and L′; as before, a normalizes the intersection H ∩ aHa = G_{P L} ∩ G_{P L′} which we will write as G_{P LL′}. Now z_{P} and z_{b(R)} commute with z_{X} of L (the group K of [AMM91] is elementary), so that bG_{P Q}b lies in G_{P LL′}, which is normalized by a. And it is on this intersection G_{P LL′} that a-action must fix the restriction of α because of A-stability. So on further conjugating by a, we see it suffices to show abg_{4}ba fixes Res_{abG_{P Q}ba}(α).

However, this last step has conjugated the original G_{P Q} to the group ⟨z_{P} , z_{X}⟩ corresponding to the points of the line L. In particular, the action of abg_{4}ba on abG_{P Q}ba interchanging z_{P} and z_{X} is induced by an involution b′ ∈ B; and B-stability of α means that b′-action fixes the restriction of α to G_{P XL}, which certainly contains z_{P} and z_{X}. So we have established the sufficient condition, yielding that g_{4}-action fixes the original Res_{G_{P Q}}(α).

(One can translate back to the level of A: an expression for U_{g_{4}} in terms of T̃_{A} and T_{g_{4}} much like that in 2.3 will show U_{b} stabilizes T̃(α), completing the proof as in 2.2).

Final remarks

The study of geometries for sporadic groups has led to the determination of quite a few suborbit diagrams (of course with respect to particular appropriate subgroups). In addition to the above examples, reasonably well-known cases include at least J_{2}, U_{4}(3) (not really a sporadic group, but it has a “sporadic” 2-local geometry), Suz, and Co.1; and potentially some much larger examples. Mod-2 cohomology has already been computed for the first three of these by Adem-Milgram and others; the methods here may allow for some streamlining of the existing proofs, and also make a start towards the larger groups whose cohomology is not yet known.

Of course “make a start” is the operative phrase, since establishing Lemma 2.1 provides a convenient approach to stable cohomology via larger subgroups, but does not replace the detailed calculation of Sylow-group cohomology.

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