INTEGRAL COHOMOLOGY RING OF U_{3}(5) AND HS AT THE PRIME 5

MATTHEW BARDOE, STEPHEN SIEGEL, AND KRISTIN UMLAND

Abstract. In this paper we calculate the localization at the prime 5 of the integral cohomology ring of the 3 dimensional Unitary group over the field of twenty-five elements, and for the Higman-Sims sporadic simple group. For both of these groups the Sylow subgroup is isomorphic to the extraspecial group of height 5 and order 125.

1. Introduction

In this paper we present the 5-part of the integral cohomology ring of the three dimensional unitary group over the field of twenty five elements, U_{3}(5). The even part of the cohomology has already been computed by Tezuka and Yagita [TY96, Thm 5.9]. In their paper Tezuka and Yagita describe the elements of the even dimensional cohomology for U_{3}(5) and HS, as well as many other groups. They also describe the relationship between the 5-part of the integral cohomology of U_{3}(5) and the 5-part of the integral cohomology of several sporadic simple groups including HS, Mcl, Co_{3} and Co_{2}.

In this paper we describe the entire cohomology ring. Also, our description of the ring is much more specific than that of Tezuka and Yagita. We give Poincaré series and module structure for both rings, and from our calculation it is easy to see how each of these cohomology rings relates to the other, as well as to the cohomology ring of their Sylow 5-subgroup.

The Sylow 5-subgroups of both U_{3}(5) and HS are isomorphic to the extraspecial group of height 5 and order 125, which we will denote 5_{+}^{1+2}. This group is probably most easily represented as the group of strictly upper triangular matrices inside SL_{3}(5).

We list our main results below.

Theorem 1.1. The ring H^{∗}(U_{3}(5), ℤ)_{(5)} is generated by:

Generator  a, b x, y, z m, n p q c r θ
Degree  5  8  14 16 24 25 34 40
Additive Order 5  5  5 5 5 5 5 25

Relations

{a, b, c} · {a, b, c} = 0 {a, b, c, m, n, p, q, r, x, y, z} · {p, q} = 0
ay = 3bz = −bx  by = az + ax  y^{2} = −x^{2} − xz
yz = 3xy z^{2} = 3x^{2} + 2xz
bm = an  ym = −nx  yn = mx + mz  zn = 3nx

Date: November 15, 1996.

The first and third authors were partially supported by NSA grant MDA 904-93-H-3039.

2 MATTHEW BARDOE, STEPHEN SIEGEL, AND KRISTIN UMLAND

am^{2} = 2cz + cx  an^{2} = 3cz + 3cx  bn^{2} = bm^{2} = 3cy
cm = ar  cn = br  m^{3} = xr + 2zr
m^{2}n = 3yr  mn^{2} = 3zr − 3xr  n^{3} = 3yr
mr = zθ  nr = 2yθ  cr = amθ + 3bnθ
r^{2} = m^{2}θ + 3n^{2}θ

This ring has the following module structure:

H^{∗}(U_{3}(5), ℤ)_{(5)} ≅

{ℤ̃_{5}[x]⟨1, a, b, y, z, az, m, n, mz, am, an, bn, c, m^{2}, mn, n^{2}, cy, cz, r, ar, br, yr, zr, azr⟩

⊕𝔽_{5}⟨p, q⟩} ⊗ ℤ̃_{25}[θ]

and Poincaré series

p_{U_{3}(5)} = \frac{1+2t^{5}+2t^{8}+t^{13}+2t^{14}+t^{16}+3t^{19}+t^{22}+t^{25}+3t^{28}−t^{32}+2t^{33}+t^{34}+2t^{39}+2t^{42}+t^{47}}{(1−t^{8})(1−t^{40})} Theorem 1.2. The ring H^{∗}(HS, ℤ)_{(5)} is generated by:

Generator  b x, z n p e q f g θ
Degree  5 8 14 16 19 24 28 39 40
Additive Order 5 5 5 5 5 5 5 5 25

Relations

{b, e, g} · {b, e, g} = 0 {b, e, f, g, n, p, q, x, z} · {p, q} = 0
bz = 3bx z^{2} = 3x^{2} + 2xz zn = 3nx
zn^{3} = 3xn^{3} n^{4} = −x^{2}θ − xzθ
bn^{3} = 3gz + 3gx gn = 3bx f^{2} = x^{2}θ
ef = gx + 2gz
bf = bn^{2} ez = gf = −ex − bnx fz = −fx − n^{2}x
en = bn^{2} nf = n^{3}

The ring has the following module structure:

H^{∗}(HS, ℤ)_{(5)} ≅ {ℤ̃_{5}[x]⟨1, b, z, n, e, bn, f, n^{2}, bn^{2}, g, n^{3}, gz⟩ ⊕ 𝔽_{5}⟨p, q⟩} ⊗ ℤ̃_{25}[θ] and Poincaré series

p_{HS} = \frac{1 + t^{5} + t^{8} + t^{14} + 2t^{19} + t^{26} + 2t^{28} − t^{32} + t^{33} + t^{39} + t^{42} + t^{47}}{(1 − t^{8})(1 − t^{40})}

2. Preliminaries

For integral cohomology it is convenient to define a functor which is a variation on the symmetric algebra. This is because most elements in our cohomology ring have finite additive order with the exception of the identity element. Let q be a positive integer and V = ⨁_{j∈ℤ} V^{j} a graded-free ℤ-module, that is, a graded ℤ-module in which each homogeneous component V^{j} is free. The tensor algebra

T (V) = ℤ ⊕ V ⊕ V^{⊗2} ⊕ ⋯

is then a graded ring, in which the homogeneous element x_{1} ⊗ ⋯ ⊗ x_{n}, where x_{i} ∈ V^{d(i)}, has degree ∑_{i=1}^{n} d(i). The ideal I generated by all qx and x⊗y −(−1)^{ij}y ⊗x (x ∈ V^{i}, y ∈ V^{j}) is homogeneous. Let S_{q}(V) = T (V)/I. Clearly S_{q} is a functor from the category of graded-free ℤ-modules to the category of graded-commutative rings.

Define

ℤ̃_{q}[x_{1}, …, x_{n}] = S_{q}(⨁_{i} ℤx_{i}).

Given a group G and a subgroup H, the G-stable elements z ∈ H^{∗}(H, ℤ)_{(p)} are those such that

Res_{H∩gHg^{−1}}^{H}(z) = g · Res_{g^{−1}Hg}^{H}(z)

for all g ∈ G. If H is a Sylow p-subgroup of G, it is well know [Bro82, III.10.3] that the restriction map

Res^{∗} : H^{∗}(G, ℤ)_{(p)} → H^{∗}(H, ℤ)_{(p)}

is an isomorphism onto the G-stable elements of H^{∗}(H, ℤ)_{(p)}. Note that at a minimum, this implies that the image of the restriction map lies in the invariant subring

H^{∗}(H, ℤ)_{(p)}^{W_{G}(H)}

where W_{G}(H) := N_{G}(H)/H as H acts trivially on H^{∗}(H, ℤ)_{(p)}. We call W_{G}(H) the Weyl group of H in G

Proposition 2.1. The image of the restriction map from the cohomology of U_{3}(5) to the cohomology of 5_{+}^{1+2} is exactly

H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}^{W_{U_{3}(5)}(5_{+}^{1+2})}.

Proof. We take G = U_{3}(5) and H ≅ 5_{+}^{1+2}, which can be identified as the strictly upper triangular matrices in G.

In order to recognize the image of the restriction map, we must analyze the restrictions from H to intersections with its conjugates. However, in this case if g ∉ N_{G}(H) then H∩gHg^{−1} is trivial. To see this first note that G acts 2-transitively on conjugates of H [C^{+}85, p. 34], so there are only two orbits of conjugates, namely {H} and the set of its non-trivial conjugates. Thus

H ∩ gHg^{−1} = H ∩ g′Hg′^{−1} ∀g, g′ ∉ N_{G}(H).

Now take the lower triangular matrices in G to be a non-trivial conjugate of H, call this subgroup H′. Since H ∩ H′ is the trivial subgroup, this completes the proof.

Thus it remains to calculate the invariants in H^{∗}(H, ℤ)_{(5)} under the action of W_{G}(H) ≅ ℤ/8.

3. The ring H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}

We start with the cohomology of H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} as presented in [Lea91, Thm. 3].

Generator  α, β μ, ν χ_{2} χ_{3} χ_{4} ζ
Degree 2 3 4 6 8 10
Additive Order 5  5 5 5 5 25

with relations

(A_{1}) αμ = βν  (A_{2}) α^{5}β = β^{5}α  (A_{3}) α^{5}μ = β^{5}ν
(B) αχ_{2} = αχ_{3} = βχ_{2} = βχ_{3} = 0
μχ_{2} = μχ_{3} = νχ_{2} = νχ_{3} = 0
(C) αχ_{4} = −α^{5}, βχ_{4} = −β^{5}, μχ_{4} = −β^{4}μ, νχ_{4} = −α^{4}ν

(D_{1}) χ_{i}χ_{j} = 0 unless i = j = 4 (D_{2}) χ_{4}^{2} = α^{8} + β^{8} − α^{4}β^{4}

(E) μν = λχ_{3} λ ∈ 𝔽_{5}^{∗}

Note that (B), (C) above allow us to express every element as a sum of polynomials in ζ, χ_{i} and polynomials in ζ, α, β, μ, ν.

Lemma 3.1. A module representation of H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} is given by:

H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} = {ℤ̃_{5}[α]⟨1, β, β^{2}, β^{3}, β^{4}, ν, μ, μβ, μβ^{2}, μβ^{3}⟩
⊕ℤ̃_{5}[β]⟨β^{5}, β^{4}μ⟩ ⊕ 𝔽_{5}⟨χ_{2}, χ_{3}, χ_{4}⟩} ⊗ ℤ̃_{25}[ζ]

Proof. As ζ is not involved in any of the relations the tensor product is clear, therefore we can restrict our attention to monomials without ζ. Note also that with the exception of relation (D_{2}), all relations are equivalences between monomials. Therefore we deal with χ_{4} first. Relation (D_{2}) shows that χ_{4}^{2} is in the span of the first two direct summands of the module structure. The relations (C) show that multiplying α, β, μ, ν by χ_{4} is contained in

ℤ̃_{5}[α](1, β, β^{2}, β^{3}, β^{4}, ν, μ, μβ, μβ^{2}, μβ^{3}) ⊕ ℤ̃_{5}[β](β^{5}, β^{4}μ).

Note that this also shows that χ_{4}^{i} is in the module structure. Relations (D_{1}) and (B) show that any multiple of χ_{2} and χ_{3} with any-other generator are zero. Therefore we have accounted for every polynomial containing any of the χ_{i} in our module structure.

We now focus on the subring generated by α, β, μ, ν. As stated above we have the following relations derived from the relations given:

(A_{2}) ⇒ α^{m}β^{n} = α^{m+4k}β^{n−4k} where 1 ≤ n − 4k ≤ 4

Therefore we have every monomial of the form α^{m}β^{n}.

We have every monomial of the form β^{m}ν and α^{m}β^{n}μ from the relations:

(A_{1}), (A_{2}) ⇒ β^{m}ν = β^{m−1}αμ = α^{1+4k}β^{m−1−4k}μ, k as above

The remaining monomials are α^{m}ν and β^{m}μ and these are clearly contained in our module structure.

Furthermore, the Poincaré series based on this decomposition agrees with the Poincaré series for H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} (this may be determined from [Sie96, Thm. 7]):

p_{5_{+}^{1+2}}(t) = \frac{1 − t + 2t^{2} + 2t^{4} − t^{5} + 2t^{6} − t^{7} + 2t^{8} − t^{9} + t^{10}}{(1 − t)(1 − t^{10})}

Using this module representation, we will determine the invariants under the Weyl group action.

4. Calculation of Invariants

The action under a fixed generator, z, of the Weyl group, W_{U_{3}(5)}(5_{+}^{1+2}) on H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} may be determined from [Lea91, Thm. 3]:

α ⟼ β β ⟼ 2α
μ ⟼ ν ν ⟼ 3μ
χ_{2} ⟼ 4χ_{2} χ_{3} ⟼ 2χ_{3}
χ_{4} ⟼ χ_{4} ζ ⟼ −7ζ

As the Weyl group is a cyclic group, its characters are all linear. We will write its 5-modular characters as follows: let σ_{i}(z) = 2^{i/2}, where z is the distinguished generator described above. As 2 is not a square in 𝔽_{5} it is clear that this generates a group of order 8 in 𝔽_{25}^{∗}.

Let H be the subring of H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} generated by everything but ζ. Since the action on ζ is −7ζ (or as 3 when multiplied by something which has additive order 5), and -7 is one of the fourth roots of unity modulo 5, invariants will be of the form

η_{i}ζ, γ_{j}^{2}ζ^{2}, δ_{k}ζ^{3}, ϵ_{l}ζ^{4}

where η_{i}, γ_{j}, δ_{k}, ϵ_{l} ∈ H and

η_{i} ⟶ 2η_{i}  γ_{j} ⟶ −γ_{j}  δ_{k} ⟶ 3δ_{k}  ϵ_{l} ⟶ ϵ_{l}

under the action of of z. Let M_{i} be the subset of H generated by all elements, x, such that the action of z is given by

x^{z} = σ_{i}(z)x

. From the above we have the following decomposition of H^{∗}(5_{+}^{1+2} :8, ℤ)_{(5)}:

H^{∗}(5_{+}^{1+2} :8, ℤ)_{(5)} = {M_{0} ⊕ M_{2}ζ ⊕ M_{4}ζ^{2} ⊕ M_{6}ζ^{3}} ⊗ ℤ̃_{25}[ζ^{4}]. (4.1)

So now we focus our attention on H. From the structure above, we can begin by finding the M_{i}’s. Note that M_{0} is a subring of H. In Table 1, we list a basis and the character for the Weyl group acting on H (up to degree 18).

Lemma 4.2. For degree greater than nine H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} ≅ H^{∗+8}(5_{+}^{1+2}, ℤ)_{(}5) as ℤ/8 modules.

Table 1. Characters up to Degree 18

Degree Dimension  Basis  Character
0  1  1  σ_{0}
1  0
2  2  α, β  σ_{1} + σ_{5}
3  2  μ, ν  σ_{3} + σ_{7}
4  4  α^{2}, αβ, β^{2}, χ_{2}  2σ_{2} + σ_{4} + σ_{6}
5  3  αμ, βμ, αν  2σ_{0} + σ_{4}
6  5  α^{3}, α^{2}β, αβ^{2}, β^{3}, χ_{3}  σ_{2} + 2σ_{3} + 2σ_{7}
7  4  α^{2}μ, αβμ, β^{2}μ, α^{2}ν  2σ_{1} + 2σ_{5}
8  6  α^{4}, α^{3}β, α^{2}β^{2}, αβ^{3}, β^{4}, χ_{4}  3σ_{0} + 3σ_{4}
9  5  α^{3}μ, α^{2}βμ, αβ^{2}μ, β^{3}μ, α^{3}ν  3σ_{2} + 2σ_{6}
10  6  α^{5}, α^{4}β, α^{3}β^{2}, α^{2}β^{3}, αβ^{4}, β^{5}  3σ_{1} + 3σ_{5}
11  6  α^{4}μ, α^{3}βμ, α^{2}β^{2}μ, αβ^{3}μ, α^{4}ν, β^{4}μ  3σ_{3} + 3σ_{7}
12  6  α^{6}, α^{5}β, α^{4}β^{2}, α^{3}β^{3}, α^{2}β^{4}, β^{6}  3σ_{2} + 3σ_{6}
13  6  α^{5}μ, α^{4}βμ, α^{3}β^{2}μ, α^{2}β^{3}μ, α^{5}ν, β^{5}μ  3σ_{0} + 3σ_{4}
14  6  α^{7}, α^{6}β, α^{5}β^{2}, α^{4}β^{3}, α^{3}β^{4}, β^{7}  3σ_{3} + 3σ_{7}
15  6  α^{6}μ, α^{5}βμ, α^{4}β^{2}μ, α^{3}β^{3}μ, α^{6}ν, β^{6}μ  3σ_{1} + 3σ_{5}
16  6  α^{8}, α^{7}β, α^{6}β^{2}, α^{5}β^{3}, α^{4}β^{4}, β^{8}  3σ_{0} + 3σ_{4}
17  6  α^{7}μ, α^{6}βμ, α^{5}β^{2}μ, α^{4}β^{3}μ, α^{7}ν, β^{7}μ  3σ_{2} + 3σ_{6}
18  6  α^{9}, α^{8}β, α^{7}β^{2}, α^{6}β^{3}, α^{5}β^{4}, β^{9}  3σ_{1} + 3σ_{5}

Proof. Note that as χ_{4} is invariant under the action of the Weyl group, multiplication by χ_{4} is a module homomorphism. The action of multiplication by χ_{4} is given by the relations χ_{4}α = −α^{5} and χ_{4}β = −β^{5}. Therefore by the module structure given in Lemma 3.1 this homomorphism is an isomorphism.

Notice that using the above result and Table 1 we are able to compute the Poincaré series for the M_{i} and H^{∗}(U_{3}(5), ℤ)_{(5)}.

Lemma 4.3. The Poincaré series for M_{0}, M_{2}, M_{4}, M_{6} and H^{∗}(U_{3}(5), ℤ)_{(5)} are, respectively:

p_{M_{0}}(t) = \frac{1 + 2t^{5} + 2t^{8} + t^{13}}{(1 − t^{8})}

p_{M_{2}}(t) = \frac{2t^{4} + t^{6} + 3t^{9} + t^{12} − t^{14}}{1 − t^{8}} = \frac{2t^{4} + 3t^{9} + t^{12}}{1 − t^{8}} + t^{6}

p_{M_{4}}(t) = \frac{t^{4} + t^{5} + 3t^{8} − t^{12} + 2t^{13}}{1 − t^{8}} = \frac{t^{5} + 3t^{8} + 2t^{13}}{1 − t^{8}} + t^{4}

p_{M_{6}}(t) = \frac{t^{4} + 2t^{9} + 2t^{12} + t^{17}}{1 − t^{8}} = t^{4} · p_{M_{0}}(t)

p_{U_{3}(5)} =\frac{1+2t^{5}+2t^{8}+t^{13}+2t^{14}+t^{16}+3t^{19}+t^{22}+t^{25}+3t^{28}−t^{32}+2t^{33}+t^{34}+2t^{39}+2t^{42}+t^{47}}{(1−t^{8})(1−t^{40})}

Proof. The Poincaré series for the M_{i} can be easily determined from Table 1 and Lemma 4.2. For p_{U_{3}(5)} we have by Equation 4.1:

p_{U_{3}(5)} = (p_{M_{0}}(t) + p_{M_{2}}(t)t^{10} + p_{M_{4}}(t)t^{20} + p_{M_{6}}(t)t^{30})/(1 − t^{40})

We determine generators and a module structure for each of the M_{i}. We start with M_{0}.

Lemma 4.4. The subring M_{0} ⊂ H ⊂ H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} is generated by:

Name  A  B X  Y  Z
In H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} αν + 3βμ αμ χ_{4} αβ^{3} + 3α^{3}β α^{4} + β^{4}
Degree  5  5 8  8  8
Additive Order  5  5 5  5  5

This ring has module structure:

M_{0} = ℤ̃_{5}[X]⟨1, A, B, Z, Y, AZ⟩

Proof. From Lemma 4.2 we know that inside of H we have the following decomposition of M_{0}:

M_{0} = M_{0} ∩ H1–9 ⊕ {M_{0} ∩ H10–17} ⊗ ℤ̃_{5}[χ_{4}]

From the Poincaré series we know that we need only find the eigenvectors of eigenvalue 1 in degrees 0 to 17.

Eigenvectors for σ_{0}

Degree  Generators
0  1
5  A = αν + 3βμ
B = αμ
X = χ_{4}
8  Y = αβ^{3} + 3α^{3}β
Z = α^{4} + β^{4}
AX = −α^{5}ν − 3β^{4}μ
13  BX = −α^{5}μ
AZ = α^{5}ν + 3β^{5}μ + α^{2}β^{3}μ + 3α^{4}βμ

X^{2} = α^{8} + β^{8} − α^{4}β^{4}

16  XY = −α^{5}β^{3} − 3α^{7}β
XZ = −α^{8} − β^{8}

These are linearly independent by the module structure given in Lemma 3.1.

Note that M_{0} is a subring of H and additionally that M_{0}, M_{2}, M_{4} and M_{6} are all M_{0}-modules.

Lemma 4.5. The modules M_{2}, M_{4}, M_{6} ⊂ H ⊂ H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} have the following structure as M_{0}-modules:

Generators for M_{2}

Name  M  N P
In H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} α^{2} + 3β^{2} αβ χ_{3}
Degree  4  4 3
Additive Order  5  5 5

Module Structure for M_{2}

M_{2} = ℤ̃_{5}[X]⟨M, N, AM, AN, BN, MZ⟩ ⊕ 𝔽_{5}⟨P ⟩

Generators for M_{4}

Name  Q  C  M^{2}  N^{2}  MN
In H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} χ_{2} αν + 2βμ α^{4} − β^{4} + α^{2}β^{2} α^{2}β^{2} α^{3}β + 3αβ^{3}
Degree  4  5  8  8  8
Additive Order  5  5  5  5  5

Module Structure for M_{4}

M_{4} = ℤ̃_{5}[X]⟨C, M^{2}, N^{2}, MN, CY, CZ⟩ ⊕ 𝔽_{5}⟨Q⟩

Generators for M_{6}

Name  R
In H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} α^{2} + 2β^{2}
Degree  4
Additive Order  5

Module Structure for M_{6}

M_{6} = ℤ̃_{5}[X]⟨R, AR, BR, Y R, ZR, AZR⟩

Proof. From Lemma 4.2 we know that inside of H that we have the following decomposition for each of the M_{i}, i ∈ {2, 4, 6}:

M_{i} = M_{i} ∩ H1–9 ⊕ {M_{i} ∩ H10–17} ⊗ ℤ̃_{5}[χ_{4}]

From the Poincaré series we need only find the eigenvectors of proper eigenvalue in degree’s zero to 17.

Eigenvectors for σ_{2}

Degree  Generators
4  M = α^{2} + 3β^{2}
N = αβ
6  P = χ_{3}

AM = α^{3}ν − β^{3}μ + α^{2}βμ

9  AN = α^{3}μ + 3αβ^{2}μ
BN = α^{2}βμ
MX = −α^{6} − 3β^{6}
12  NX = −α^{5}β

MZ = α^{6} + 3β^{6} + α^{2}β^{4} + 3α^{4}β^{2} AMX = −α^{7}ν + β^{7}μ − α^{6}βμ

17  ANX = −α^{7}μ − 3α^{5}β^{2}μβ
BNX = −α^{6}βμ

Eigenvectors for σ_{4}

Degree  Generators
4  Q = χ_{2}
5  C = αν + 2βμ

M^{2} = α^{4} − β^{4} + α^{2}β^{2}

8  N^{2} = α^{2}β^{2}
MN = α^{3}β + 3αβ^{3}
CX = −α^{5}ν − 2β^{5}μ
13  CY = 2α^{3}β^{2}μ

CZ = α^{5}ν + 2β^{5}μ + α^{2}β^{3}μ + 2α^{4}βμ M^{2}X = −α^{8} + β^{8} − α^{6}β^{2}

16  N^{2}X = −α^{6}β^{2}

MNX = −α^{7}β − 3α^{5}β^{3}

Eigenvectors for σ_{6}

Degree  Generators
4  R = α^{2} + 2β^{2}
9  AR = α^{3}ν + β^{3}μ
BR = α^{3}μ + 2αβ^{2}μ
XR = −α^{6} − 2β^{6}
12  Y R = 2α^{3}β^{3}

ZR = α^{6} + 2β^{6} + α^{2}β^{4} + 2α^{4}β^{2}

AXR = −α^{7}ν − β^{7}μ
17  BXR = −α^{7}μ − 2α^{5}β^{2}μ

AZR = α^{7}ν + β^{7}μ + 2α^{4}β^{3}μ

These are linearly independent by the module structure given in Lemma 3.1.

5. Cohomology of U_{3}(5)

Having studied the stability conditions, and found the invariant inside of the cohomology ring, H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}, we are ready to put these considerations together to prove Theorem 1.1.

Proof of Theorem 1.1. From Proposition 2.1 we know that

H^{∗}(U_{3}(5), ℤ)_{(5)} = H^{∗}(5_{+}^{1+2} :8, ℤ)_{(5)}.

By Equation 4.1 and Lemmas 4.4 & 4.5, we have the following generators and module structure for H^{∗}(U_{3}(5), ℤ)_{(5)}.

H^{∗}(U_{3}(5), ℤ)_{(5)} =

{ℤ̃_{5}[x]⟨1, a, b, z, y, az, m, n, mz, am, an, bn, c, m^{2}, mn, n^{2}, cy, cz, r, ar, br, xr, yr, azr⟩

⊕𝔽_{5}⟨p, q⟩} ⊗ ℤ̃_{25}[θ]

By Lemma 4.3 we have Poincaré series:

p_{U_{3}(5)} = \frac{1+2t^{5}+2t^{8}+t^{13}+2t^{14}+t^{16}+3t^{19}+t^{22}+t^{25}+3t^{28}−t^{32}+2t^{33}+t^{34}+2t^{39}+2t^{42}+t^{47}}{(1−t^{8})(1−t^{40})}

What remains to prove is the that the relations we have listed in Theorem 1.1 really do present the ring in question. We list the relations and module structure for H^{∗}(U_{3}(5), ℤ)_{(5)} again for readability.

Table 2. Generators for H^{∗}(U_{3}(5), ℤ)_{(5)}

Name Generator (written in H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}) Degree AdditiveOrder

a  αν + 3βμ  5  5
b  αμ  5  5
x  χ_{4}  8  5
y  αβ^{3} + 3α^{3}β  8  5
z  α^{4} + β^{4}  8  5
m  α^{2}ζ + 3β^{2}ζ  14  5
n  αβζ  14  5
p  χ_{3}ζ  16  5
q  χ_{2}ζ^{2}  24  5
c  ανζ^{2} + 2βμζ^{2}  25  5
r  α^{2}ζ^{3} + 2β^{2}ζ^{3}  34  5
θ  ζ^{4}  40  25

Relations for H^{∗}(U_{3}(5), ℤ)_{(5)}

{a, b, c} · {a, b, c} = 0 {a, b, c, m, n, p, q, r, x, y, z} · {p, q} = 0

ay = 3bz = −bx  by = az + ax  y^{2} = −x^{2} − xz
yz = 3xy z^{2} = 3x^{2} + 2xz
bm = an  ym = −nx  yn = mx + mz  zn = 3nx
am^{2} = 2cz + cx  an^{2} = 3cz + 3cx  bn^{2} = bm^{2} = 3cy
cm = ar  cn = br  m^{3} = xr + 2zr
m^{2}n = 3yr  mn^{2} = 3zr − 3xr  n^{3} = 3yr
mr = zθ  nr = 2yθ  cr = amθ + 3bnθ
r^{2} = m^{2}θ + 3n^{2}θ
Module structure for iH^{∗}(U_{3}(5), ℤ)_{(5)}
H^{∗}(U_{3}(5), ℤ)_{(5)} ≅

{ℤ̃_{5}[x]⟨1, a, b, y, z, az, m, n, mz, am, an, bn, c, m^{2}, mn, n^{2}, cy, cz, r, ar, br, yr, zr, azr⟩ ⊕𝔽_{5}⟨p, q⟩} ⊗ ℤ̃_{25}[θ]

It is not too difficult to check that the relations we have given are true inside the H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} using the description for the generators we have given above. Therefore we know that the abstract ring with the presentation given in Theorem 1.1 has as a homomorphic image H^{∗}(U_{3}(5), ℤ)_{(5)}. It will be enough then to show that, using the relations given, any monomial is equivalent to something in the module structure given above. This will say that the abstract ring is no bigger than the cohomology ring as described inside of H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}.

We start with monomials formed from products of a, b, x, y, z and θ. As the terms a^{2}, ab, b^{2} are all zero we note that any monomial must be of the form ay^{∗}z^{∗}x^{∗}θ^{∗}, by^{∗}z^{∗}x^{∗}θ^{∗}, or y^{∗}z^{∗}x^{∗}θ^{∗}. In any of these cases, using the relations for y^{2}, yz, z^{2}, we see that we may rewrite these monomials as sums of monomials of the form ayx^{∗}θ^{∗}, azx^{∗}θ^{∗}, byx^{∗}θ^{∗}, bzx^{∗}θ^{∗}, yx^{∗}θ^{∗}, zx^{∗}θ^{∗} plus something in the module structure; this follows from an easy induction argument on the sum of the exponents of y and z. Then applying the relations for ay, bz, and by, we reduce to monomials of the form ax^{∗}θ^{∗}, bx^{∗}θ^{∗}, x^{∗}θ^{∗}, yx^{∗}θ^{∗}, zx^{∗}θ^{∗}, θ^{∗}, and azx^{∗}θ^{∗}, which are all contained in the module structure. In the process of rewriting we get terms that are written only with a, b, x, y, and z. This shows that these generate a subring, which we denote by m_{0}. Note that anything of the form a^{∗}b^{∗}x^{∗}y^{∗}z^{∗} may be reinterpreted into something in the module structure above.

We look at the action by multiplication of m_{0} on the generators m and n. For the generators m, n we have relations showing how to take anything of the form x^{∗}y^{∗}z^{∗}m or x^{∗}y^{∗}z^{∗}n to a sum of things of the form x^{∗}m, x^{∗}n, and x^{∗}zm. As bm = an, we can rewrite anything of the from a^{∗}b^{∗}x^{∗}y^{∗}z^{∗}m or a^{∗}b^{∗}x^{∗}y^{∗}z^{∗}n into the sum of things of the form x^{∗}m, x^{∗}n, x^{∗}zm, ax^{∗}m, ax^{∗}n, ax^{∗}zm, and bx^{∗}n. Therefore if we can translate something in the form ax^{∗}zm into the module structure we will have shown the m_{0} module generated by m and n is completely represented in the module structure. We have the relation yn = mx + mz, so ayn = amx + amz and using ay = −bx, we have amz = −amx − bnx. This shows that ax^{∗}zm may be rewritten into the terms of the module structure.

We apply the same kind of analysis to the module generated by m^{2}, mn, n^{2}, and c. As cx, cy, and cz, are all in the module structure then using the relations described above, any term of the form cx^{∗}y^{∗}z^{∗} may be rewritten as a the sum of things of the form cx^{∗}, cx^{∗}y, cx^{∗}z, and as ac = 0 and bc = 0 we see that anything of the form a^{∗}b^{∗}x^{∗}y^{∗}z^{∗}c may be reinterpreted into something in the module structure. Multiplication by terms of the form x^{∗}y^{∗}z^{∗} are not a problem by the last paragraph. Therefore we only need understand am^{2}, amn, an^{2} and bm^{2}, bmn, bn^{2} in terms of the module structure. Note that there are relations for am^{2}, an^{2}, bm^{2} and bn^{2} and that amn = bm^{2} and an^{2} = bmn. Therefore the m_{0} module generated by m^{2}, mn, n^{2} and c is completely represented in terms of the module structure.

We look at the m_{0} module generated by r. This is quite easy though, as one can immediately see that this module is isomorphic to m_{0} as an m_{0} module and therefore that everything may be interpreted in the module structure.

Finally we look at products from the different modules defined above. If the product between any two of these generators is able to be reinterpreted into the module structure we will be done. One easily sees that this is done for all the relevant products of the set m, n, m^{2}, n^{2}, mn, c, r.

6. Cohomology of HS

The Higman-Sims group is one of the 26 sporadic simple groups, and contains in it a maximal subgroup isomorphic to U_{3}(5) : 2. This subgroup has index relatively prime to 5, therefore the restriction map from H^{∗}(HS, ℤ)_{(5)} to H^{∗}(U_{3}(5) : 2, ℤ)_{(5)} is injective. The following result is also obtained in [TY96, Thm. 5.9].

Lemma 6.1. The image of the restriction map from the 5-local cohomology of HS to the 5-local cohomology of U_{3}(5):2 is exactly:

H^{∗}(U_{3}(5):2, ℤ)_{(5)} = H^{∗}(U_{3}(5), ℤ)_{(5)}^{W_{HS}(U_{3}(5))}

Proof. The permutation representation defined by this maximal subgroup gives a 2-transitive action as can be seen from the character for this representation listed in the [C^{+}85, p. 80]. It can be easily calculated that that the intersection of any two conjugate U_{3}(5):2 subgroups have intersection which has order relatively prime to 5. Therefore the 5-local cohomology ring of the intersection of these two subgroups is trivial. This proves the result.

Therefore the cohomology of HS is given by the invariants under the action of a single involution on the cohomology of U_{3}(5). In [TY96, p. 501] Tezuka and Yagita give a description of N_{HS}(5_{+}^{1+2}). From this and [Lea91, Thm. 3] we can describe the action of this element of order 2 on H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}. Since we have identified H^{∗}(U_{3}(5), ℤ)_{(5)} inside of H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}, we may use this description to obtain the action of the involution on H^{∗}(U_{3}(5), ℤ)_{(5)}.

The action of the involution on the generators of H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} is given by:

α ⟼ 3α β ⟼ 2β
μ ⟼ 2μ ν ⟼ 3ν
χ_{2} ⟼ χ_{2} χ_{3} ⟼ χ_{3}
χ_{4} ⟼ χ_{4} ζ ⟼ ζ

Proof of Theorem 1.2. From the action defined above and Table 2 we are able to compute the action of this involution on the generators of H^{∗}(U_{3}(5), ℤ)_{(5)}.

a ⟼ −a  b ⟼ b  x ⟼ x
y ⟼ −y  z ⟼ z  m ⟼ −m
n ⟼ n  p ⟼ p  q ⟼ q
c ⟼ −c  r ⟼ −r  θ ⟼ θ

So the generators which are not invariant are a, y, m, c, and r. Thus the invariants under the action of y are generated by I = {b, x, z, n, p, q, θ} and the products of two generators in {a, y, m, c, r}. Those products which are linearly independent of I are {am, ar, m^{2}}. We make the following renaming for ease of notation: e = am, f = m^{2}, and g = ar. We list the generators, their degrees, additive orders, and images in H^{∗}(5_{+}^{1+2}, ℤ)_{(5)} and H^{∗}(U_{3}(5), ℤ)_{(5)} in Table 3.

Table 3. Generators for H^{∗}(HS, ℤ)_{(5)}

Name  Generator  Generator  Degree Additive

written in

written in H^{∗}(5_{+}^{1+2}, ℤ)_{(5)}

order

H^{∗}(U_{3}(5), ℤ)_{(5)}

b  b  αμ  5  5
x  x  χ_{4}  8  5
z  z  α^{4} + β^{4}  8  5
n  n  αβζ  14  5
p  p  χ_{3}ζ  16  5
e  am  α^{3}νζ − β^{3}μζ + α^{2}βμζ  19  5
q  q  χ_{2}ζ^{2}  24  5
f  m^{2}  α^{4}ζ^{2} − β^{4}ζ^{2} + α^{2}β^{2}ζ^{2}  28  5
g  ar  α^{3}νζ^{3} + β^{3}μζ^{3}  39  5
θ  θ  ζ^{4}  40  25

To determine the module structure for this ring all we need to do is see which generators for the module structure for H^{∗}(U_{3}(5), ℤ)_{(5)} are invariant. This follows from the fact that that x and θ are invariant under the involution. Therefore we have the following module structure.

H^{∗}(HS, ℤ)_{(5)} ≅ {ℤ̃_{5}[x]⟨1, b, z, n, e, bn, f, n^{2}, bn^{2}, g, n^{3}, gz⟩ ⊕ 𝔽_{5}⟨p, q⟩} ⊗ ℤ̃_{25}[θ]

Again it remains to prove that the relation we have listed in Theorem 1.2 present the given ring. We list the relations and module structure again for readability.

Relations for H^{∗}(HS, ℤ)_{(5)}
{b, e, g} · {b, e, g} = 0 {b, e, f, g, n, p, q, x, z} · {p, q} = 0
bz = 3bx  z^{2} = 3x^{2} + 2xz  zn = 3nx
zn^{3} = 3xn^{3} n^{4} = −x^{2}θ − xzθ
bn^{3} = 3gz + 3gx gn = 3bx f^{2} = x^{2}θ
ef = gx + 2gz
bf = bn^{2} ez = gf = −ex − bnx fz = −fx − n^{2}x
en = bn^{2} nf = n^{3}
Module structure for H^{∗}(HS, ℤ)_{(5)}

H^{∗}(HS, ℤ)_{(5)} ≅ {ℤ̃_{5}[x]⟨1, b, z, n, e, bn, f, n^{2}, bn^{2}, g, n^{3}, gz⟩ ⊕ 𝔽_{5}⟨p, q⟩} ⊗ ℤ̃_{25}[θ]

Again we show that these relations are enough to imply that the ring is no bigger than the module structure given above. We do this in pieces, first showing that there is a subring which is no bigger than its image in the full ring, and then adding more modules for this subring. Finally we show the entire ring is presented in the module structure.

We start with the subring generated by b, x, z and θ. As we have relation for rewriting bz and z^{2} in terms of the module structure, this subring is represented in the module structure. Again as all these products are reinterpreted as monomials in b, x, z and θ these generate a subring. Let this subring be denoted by n_{0}.

We look at the n_{0} module generated by n, bn, p and e. In this case bp, xp, zp and p^{2} are all zero. Therefore every term with p in it may be written in terms of the module structure. As for terms of the form b^{∗}x^{∗}z^{∗}θ^{∗}n, using the zn = 3nx we are able to rewrite this term without factors of z, and therefore any term of this form may be rewritten in the module structure. Similarly, using the relations ez = −ex − bnx and be = 0, we are able to rewrite any term of the form b^{∗}x^{∗}z^{∗}θ^{∗}e as a sum of terms of the form bx^{∗}θ^{∗}n and x^{∗}θ^{∗}e, and therefore it is in the module structure.

The n_{0} module generated by q, n^{2}, bn^{2} and f is again contained in the module structure. For the module spanned by q this can be seen, as in the case of p above, since all relevant products are zero except qθ which is in the module structure. For the module spanned by f we again have relations which allow us to rewrite any term with both f and b or f and z in terms of the sums of terms in the module structure. This is also the case for terms with a factor of n^{2} as bn^{2} is part of the module structure and we rewrite n^{2}z as 3n^{2}x, so we may rewrite anything of the form b^{∗}x^{∗}z^{∗}θ^{∗}n^{2} as the sum of terms from the module structure.

The last n_{0} module we need look at is generated by g, n^{3} and gz. For the module generated by g, as bg = 0 and z^{2} = 3x^{2} + 2xz we may rewrite the cyclic n_{0}-module generated by g in terms of the module structure. Finally for n^{3} there are relations showing how to rewrite bn^{3} = 3gz + 3gx and zn^{3} = 3n^{3}x as above for n and n^{2}. Therefore this module is contained in the module structure.

Finally we must look at products between any two generators of these modules. Products with p or q are no problem as they are all zero. We have relations for e^{2}, en, ef, eg, f^{2}, fg, fn, g^{2}, gn, and n^{4}, which show how to rewrite these products interms of the module structure, so we are done.

7. Acknowledgments

The authors would like to thank the organizers and funders of the 1996 AMS Summer Institute in Seattle, Washington.

References

[Bro82] K. S. Brown, Cohomology of groups, Graduate texts in mathematics, vol. 66, Springer-Verlag, Berlin–New York, 1982.

[C^{+}85] J. H. Conway et al., Atlas of finite groups, Clarendon Press, Oxford, 1985.

[Lea91] I. J. Leary, The intergral cohomology rings of some p-groups, Math. Proc. Camb. Phil. Soc. 110 (1991), 25–32.

[Sie96] Stephen F. Siegel, The spectral sequence of a split extension and the cohomology of an extraspecial group of order p^{3} and exponent p, Journal of Pure and Applied Algebra 106 (1996), no. 2, 185–198.

[TY96] M. Tezuka and N. Yagita, On odd prime components of cohomologies of sporadic simple groups and the rings of universal stable elements, J. Algebra 183 (1996), 483–513.

Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan, Chicago Illinois 60607-7045 USA

   E-mail address: bardoe@math.uic.edu

Dept of Mathematics & Statistics, LGRT, Box 34515, University of Massachusetts, Amherst MA 01003-4515 USA

   E-mail address: siegel@math.umass.edu

Department of Mathematics and Statistics, University of New Mexico, Albuquerque New Mexico, 87131 USA

   E-mail address: umland@math.unm.edu