COHOMOLOGY OF THE DOUBLE COVER OF THE MATHIEU GROUP M_{12}.

D. J. BENSON AND J. F. CARLSON

Abstract. In this paper we calculate the mod two cohomology of the double cover of the Mathieu group M_{12}. The starting point is the calculation by Adem, Maginnis and Milgram of the mod two cohomology of M_{12}. We use a hypercohomology spectral sequence to determine a differential in the Lyndon–Hochschild–Serre spectral sequence of the central extension, and this gets us as far as the E_{∞} page. Ungrading requires restriction to the Sylow 2-subgroup, and here some computer calculations come to our rescue.

1. Introduction

Adem, Maginnis and Milgram [1] have calculated the mod two cohomology of the Mathieu group M_{12}. This is a sporadic simple group of order 95, 040, and further information about this group can be found by consulting [8, 9].

The universal central extension of M_{12} is the double cover 2M_{12}, and the purpose of this paper is to calculate the cohomology of this double cover. A part of the point of this is to indicate a general method of attack on cohomology of double covers. This involves a combination of the Lyndon–Hochschild–Serre spectral sequence of the central extension and a technique [2] in the cohomology of finite groups which involves a spectral sequence converging to a finite dimensional graded vector space satisfying Poincaré duality. The existence of such a spectral sequence imposes severe restrictions on the possible shape of the cohomology ring of a finite group. We use this technique for example in this paper in order to prove that the d_{4} differential in the Lyndon– Hochschild–Serre spectral sequence for the central extension is zero. Finally ungrading the E_{∞} page of the Lyndon–Hochschild–Serre spectral sequence involves recourse to some computer calculations. The programs are designed only for computing with p-groups, so this involves restricting to a Sylow 2-subgroup.

The final result of our calculations is as follows:

Theorem 1.1. H^{∗}(2M_{12}, 𝔽_{2}) is generated by y and z in degree three, β, θ and τ in degree four, γ and ξ in degree five, δ, ϕ and ρ in degree six, ε and η in degree seven, χ and ω in degree eight, and ψ in degree nine, subject to the following relations:

(1) y^{2} = 0, (2) yz = 0, (3) yγ = 0, (4) yδ = 0, (5) yε = 0, (6) γ^{2} = zε + z^{2}β,

(7) γε = z^{2}δ, (8) ε^{2} = zγδ + zβε, (9) yθ = 0, (10) yϕ = 0, (11) zϕ = γθ,

(12) γϕ = (ε + zβ)θ, (13) εϕ = zδθ, (14) yξ = 0, (15) yη = 0, (16) yχ = 0,

(17) yψ = 0, (18) zη = γξ, (19) γη = (ε + zβ)ξ, (20) εη = zδξ, (21) zχ = δξ + γδ,

Both authors are partly supported by grants from the NSF.

2 D. J. BENSON AND J. F. CARLSON

(22) zψ = εξ, (23) γψ = zδξ, (24) (ε + zβ)ψ = γδξ, (25) γχ = δη + (ε + zβ)δ,

(26) δψ = εχ + zδ^{2}, (27) yρ = 0, (28) yω = 0, (29) zω = γρ + εθ,

(30) γω = (ε + zβ)ρ + zδθ, (31) εω = zδρ + (γδ + βε)θ, (32) θ^{2} = zξ,

(33) θϕ = γξ, (34) ϕ^{2} = (ε + zβ)ξ, (35) θξ = zρ, (36) ϕξ = γρ,

(37) θη = γρ, (38) ϕη = (ε + zβ)ρ, (39) θχ = δρ + δϕ, (40) ϕχ = δω + δθβ,

(41) θψ = ερ, (42) ϕψ = εω + (γδ + βε)θ, (43) ξ^{2} = z^{2}τ + zε, (44) ξη = zγτ + γε,

(45) ξχ = zδτ + γχ + zβδ, (46) ξψ = zετ + z(γδ + βε), (47) η^{2} = z(ε + zβ)τ + zγδ,

(48) ηχ = γδτ + (ε + zβ)χ + βγδ, (49) ηψ = z^{2}δτ + zεδ, (50) χ^{2} = δ^{2}τ + βδ^{2},

(51) χψ = δετ + δ^{2}ξ + γδ^{2} + βδε, (52) ψ^{2} = z(γδ + βε) + z^{2}δ^{2} + zβγδ + zβ^{2}ε,

(53) θρ = z^{2}τ + zε, (54) ϕρ = zγτ + z^{2}δ, (55) θω = zγτ + εξ + z^{2}δ,

(56) ϕω = z(ε + zβ)τ + εη + zγδ, (57) ξρ = zθτ + εθ, (58) ηρ = γθτ + zδθ,

(59) χρ = δθτ + δω, (60) ψρ = εθτ + (γδ + βε)θ, (61) ξω = γθτ + zδθ + ερ,

(62) ηω = (ε + zβ)θτ + γδθ + zδρ, (63) χω = δϕτ + βδρ,

(64) ψω = zδθτ + (γδ + βε)ρ + δεθ, (65) ρ^{2} = zξτ + εξ,

(66) ρω = γξτ + zετ + zδξ + zγδ + zβε, (67) ω^{2} = (ε + zβ)ξτ + βεξ. The action of the Steenrod algebra is as follows:

generator degree  Sq^{1}  Sq^{2}  Sq^{4}  Sq^{8}
y  3  0  0
z  3  0  γ
β  4  0  z^{2} + δ  β^{2}
θ  4  ξ  ϕ  zξ
τ  4  0  δ  τ^{2}
γ  5  z^{2}  0  βγ + zδ + z^{3}
ξ  5  0  zτ + ε + η  γτ + zδ
δ  6  ε  0  βδ + zε
ϕ  6  zθ + η  zξ  (z^{2} + δ)θ + βϕ
ρ  6  yβ + zτ + ε  θτ + ω  δθ + τϕ + θρ
ε  7  0  0  βε
η  7  zξ  γτ + zδ  (z^{2} + δ)ξ + βη
χ  8  ψ  βδ + zε + δτ z^{2}δ + βχ + zψ  χ^{2}
ω  8  zδ + γτ  δθ + ϕτ  z^{2}ρ + δρ  ω^{2}
+zρ + ψ  +z^{2}τ + zε  +ρ^{2} + βε
ψ  9  0  yβ^{2} + βε + δξ  βψ  δ^{2}ξ + zξψ
+ετ + zχ  +χψ

The Poincaré series is

∑_{i=0}^{∞} t^{i} dim_{𝔽_{2}} H^{i}(2M_{12}, 𝔽_{2}) = \frac{1 + t^{3} + t^{4} + 2t^{5} + t^{6} + 2t^{7} + 2t^{8} + t^{12} + t^{13}}{(1 − t^{3})(1 − t^{4})^{2}(1 − t^{6})}

= 1 + 2t^{3} + 3t^{4} + 2t^{5} + 4t^{6} + 7t^{7} + 9t^{8} + 9t^{9} + 12t^{10} + 18t^{11}

+ 23t^{12} + 23t^{13} + 28t^{14} + 38t^{15} + 45t^{16} + 47t^{17} + …

In the presentation for H^{∗}(2M_{12}, 𝔽_{2}) given in the above theorem, the generators and relations have the following interpretation in terms of the spectral sequence of the central extension. The generators y, z, β, γ, δ and ϵ are inflations from H^{∗}(M_{12}, 𝔽_{2}), and relations (1)–(8) give a presentation for the image of inflation. The elements θ and ϕ are representatives of generators of row one of the spectral sequence, and relations (9)–(13) ungrade a presentation for row one as a module over the image of inflation. The elements ξ, η, χ and ψ are representatives of generators of row two of the spectral sequence, and relations (14)–(26) ungrade a presentation for row two as a module over the image of inflation. The elements ρ and ω are representatives of generators of row three of the spectral sequence, and relations (27)–(31) ungrade a presentation for row three as a module over the image of inflation. The element τ is a regular element representing a degree four generator on the vertical axis. Multiplication by this element gives the vertical periodicity in the spectral sequence. Finally, relations (32)–(67) ungrade the relations given by expressing 36 products among the eight generators on rows one, two and three in terms of the generators on the appropriate rows.

The Sylow 2-subgroup P of M_{12} has five conjugacy classes of maximal elementary abelian 2-subgroups, with representatives E_{i}, 1 ≤ i ≤ 5. In M_{12}, E_{1} is conjugate to E_{2} and E_{3} is conjugate to E_{4}, leaving three conjugacy classes of maximal elementary abelian 2-subgroups. These detect cohomology of M_{12}. The inverse images Ê_{i} in 2M_{12} of the subgroups E_{i} of M_{12} detect mod two cohomology. In the Appendix, we give a presentation for the Sylow 2-subgroup P̂ of 2M_{12}, we describe the subgroups Ê_{i} with respect to this presentation, and we give the restriction maps to these subgroups from 2M_{12} as well as from P̂.

The group 2M_{12} has a faithful irreducible 12-dimensional real orthogonal representation χ_{12}, whose Stiefel–Whitney classes are given as follows:

w_{1}(χ_{12}) = 0, w_{2}(χ_{12}) = 0, w_{4}(χ_{12}) = τ + β, w_{8}(χ_{12}) = τ^{2} + zξ = (τ + θ)^{2}.

The only other faithful orthogonal representation that might be of interest for providing Stiefel–Whitney classes has dimension 20, and is the sum of two 10-dimensional complex irreducibles. It turns out that this representation has the same w_{1}, w_{2}, w_{4} and w_{8} as χ_{12}. We have not calculated w_{16} for this representation, but since the generators for the cohomology of 2M_{12} all lie in degree at most 9, it is unlikely to give an interesting interpretation for a generator.

2. Cohomology of M_{12}

We first recall from pp. 120–121 of Adem, Maginnis and Milgram [1] the mod two cohomology of M_{12}. However, rather than simply copying their presentation, we have made a change of basis in such a way as to make the presentation simpler. The following table explains the relationship between our names and those of [1].

generator degree name in notation of [1]

Sq^{1}

Sq^{2}

Sq^{4}
α  2  α  x  α^{2}
x  3  x  0  αx
y  3  y  α^{2} αx + αy
z  3  x + y + z  0  γ
β  4  β + α^{2}  0 δ + αβ + z^{2}  β^{2}
γ  5  γ + αy  z^{2}  0  βγ + zδ + z^{3}
δ  6  δ + x^{2} + α^{3} + αβ + z^{2} ε  αδ  βδ + zε
ε  7  ε + xβ + α^{2}x  0  0  βε + α^{2}ε

In terms of this new basis, the cohomology ring is given as follows: Theorem 2.1. The ring H^{∗}(M_{12}, 𝔽_{2}) is generated by elements α, x, y, z, β, γ, δ and ε (with degrees as above), and the following is a complete set of relations^{1}:

(i) αz = 0  (ii) y^{2} = α^{3} + x^{2} + xy
(iii) xz = 0  (iv) yz = 0
(v) αγ = 0  (vi) xγ = 0
(vii) yγ = 0  (viii) xδ = 0
(ix) yδ = αε  (x) xε = 0
(xi) yε = α^{2}δ  (xii) γ^{2} = zε + z^{2}β
(xiii) γε = z^{2}δ  (xiv) ε^{2} = zγδ + zβε + αδ^{2}

The action of the Steenrod operations is given in the above table.

It follows from the above relations that every element of H^{∗}(M_{12}, 𝔽_{2}) is a linear combination of monomials of the forms:

z^{?}β^{?}γ^{0/1}δ^{?} z^{?}β^{?}δ^{?}ε^{0/1} α^{?}β^{?}δ^{?}ε^{0/1} α^{?}x^{?}y^{0/1}β^{?}

where ? denotes an arbitrary exponent and 0/1 denotes an exponent of zero or one. The Poincaré series for the graded vector space with the above monomials as basis is

\frac{(1 + t^{3})^{2}}{(1 − t^{2})(1 − t^{3})(1 − t^{4})}.

1Actually, relation (xiv) contradicts the formula given in [1] for ε^{2}, but seems nonetheless to be correct.

Since this agrees with the formula for the Poincaré series of H^{∗}(M_{12}, 𝔽_{2}) given in [1], it follows that the above monomials form a basis for H^{∗}(M_{12}, 𝔽_{2}).

We now determine the minimal primes and the restriction maps to maximal elementary abelian subgroups.

Lemma 2.2. The minimal primes in H^{∗}(M_{12}, 𝔽_{2}) are as follows:

a) (α, x, y), with quotient

𝔽_{2}[z, β, γ, δ, ε]/(γ^{2} + z(ε + zβ), zγδ + ε(ε + zβ), γε + z^{2}δ)

b) (x, z, γ), with quotient

𝔽_{2}[α, y, β, δ, ε]/(α^{3} + y^{2}, yδ + αε, yε + α^{2}δ, ε^{2} + αδ^{2})

c) (z, γ, δ, ε), with quotient

𝔽_{2}[α, x, y, β]/(x^{2} + xy + y^{2} + α^{3}).

Proof. If a minimal prime does not contain z, then relations (i), (iii) and (iv) imply that it contains α, x and y; the quotient by (α, x, y) is an integral domain, so this is the unique minimal prime not containing z.

If a minimal prime contains z, then relation (xii) implies that it also contains γ. Then it either contains x, in which case it is (x, z, γ) since the quotient by this ideal is an integral domain, or it does not contain x, in which case by relations (viii) and (x) it contains δ and ε. The quotient by (z, γ, δ, ε) is also an integral domain, so we are finished.

Theorem 2.3. The group M_{12} has three conjugacy classes of maximal elementary abelian 2-subgroups, each of order eight. Let E_{a}, E_{b}, E_{c} be representatives of these; then the notation may be chosen so that

H^{∗}(E_{i}, 𝔽_{2}) = 𝔽_{2}[u, v, w] (i = a, b, c)

and the restriction maps are given as follows:

a) res(α) = res(x) = res(y) = 0,

res(z) = uv(u + v)
res(β) = u^{4} + v^{4} + w^{4} + u^{2}v^{2} + u^{2}w^{2} + v^{2}w^{2} + u^{2}vw + uv^{2}w + uvw^{2}
= (u^{2} + uv + v^{2})^{2} + w(u + w)(v + w)(u + v + w)
res(γ) = uv(u + v)(u^{2} + uv + v^{2})
res(δ) = w(u + w)(v + w)(u + v + w)(u^{2} + uv + v^{2})
res(ε) = uvw(u + v)(u + w)(v + w)(u + v + w)

The normalizer in M_{12} of E_{a} is a semidirect product of the quaternion group Q_{8} by its automorphism group Σ_{4}.

b) res(x) = res(z) = res(γ) = 0,

res(α) = u^{2}
res(y) = u^{3}
res(β) = v^{4} + w^{4} + u^{2}v^{2} + u^{2}w^{2} + v^{2}w^{2} + u^{2}vw + uv^{2}w + uvw^{2}

res(δ) = vw(u + v)(u + w)(v + w)(u + v + w)
res(ε) = uvw(u + v)(u + w)(v + w)(u + v + w)

The normalizer in M_{12} of E_{b} is a semidirect product of (ℤ/4)^{2} by the dihedral group of order twelve, D_{12}.

c) res(z) = res(γ) = res(δ) = res(ε) = 0,

res(α) = u^{2} + uv + v^{2}
res(x) = uv(u + v)
res(y) = u^{3} + u^{2}v + v^{3}
res(β) = w(u + w)(v + w)(u + v + w)

The normalizer in M_{12} of E_{c} has index two in a suitable conjugate of the normalizer of E_{a}, and is a semidirect product of Q_{8} by the alternating group A_{4}.

Proof. By a theorem of Quillen [11, 12], the minimal primes are exactly the kernels of the restriction maps to representatives of the conjugacy classes of elementary abelian 2-subgroups. In order to determine the restriction maps, it suffices to find homomorphisms of algebras, commuting with Steenrod operations, and with the right kernels.

Remarks. It is easily seen from the above that H^{∗}(M_{12}, 𝔽_{2}) is detected by restrictions to elementary abelian 2-subgroups, or equivalently that there are no nilpotent elements in this cohomology ring.

The elements β+α^{2}, δ+αβ+x^{2}+z^{2} and ε+xβ generate a copy of the Dickson algebra inside H^{∗}(M_{12}, 𝔽_{2}), which restrict to the Dickson invariants inside the cohomology of each maximal elementary abelian subgroup. This presumably explains the choice of generators used in [1].

3. Spectral sequence of the central extension

In this section, we get as far as we can using the Lyndon–Hochschild–Serre spectral sequence of the central extension

E_{2}^{∗,∗} = H^{∗}(M_{12}, H^{∗}(ℤ/2, 𝔽_{2})) ⇒ H^{∗}(2M_{12}, 𝔽_{2}).

At the end of this section, we are left with a small collection of indeterminacies in the behavior of this spectral sequence. In the next section, we use the Poincaré duality method from [2] to clear up these indeterminacies and complete the proof of Theorem 1.1.

The E_{2} page of the spectral sequence of the central extension is a tensor product

E_{2}^{∗,∗} = H^{∗}(M_{12}, 𝔽_{2}) ⊗ 𝔽_{2}[t]

with E_{2}^{∗,0} = H^{∗}(M_{12}, 𝔽_{2}) and t ∈ E_{2}^{0,1}. Since the class of the group extension must be α ∈ H^{2}(M_{12}, 𝔽_{2}), we have d_{2}(t) = α. So we have

E_{3}^{∗,q} = t^{q}. \{ \begin{matrix}H^{∗}(M_{12}, 𝔽_{2})/(α) & if q is even \\ Kernel of α on H^{∗}(M_{12}, 𝔽_{2}) & if q is odd. \end{matrix}

Since transgressions commute with Steenrod operations by a theorem of Serre [13], we have

d_{3}(t^{2}) = d_{3}(Sq^{1}(t)) = Sq^{1}(d_{2}(t)) = Sq^{1}(α) = x.

Lemma 3.1. The kernel of multiplication by α on H^{∗}(M_{12}, 𝔽_{2}) is the ideal generated by z and γ. Multiplication by x is the zero map on this ideal.

The kernel of multiplication by x on H^{∗}(M_{12}, 𝔽_{2})/(α) is the ideal generated by z, β, δ and ε.

Proof. This follows directly from the presentation of H^{∗}(M_{12}, 𝔽_{2}) given in Theorem 2.1.

This means that the E_{4} page of this spectral sequence is given by

E_{4}^{∗,q} = t^{q} · \{ \begin{matrix}H^{∗}(M_{12}, 𝔽_{2})/(α, x) & q ≡ 0 (mod 4) \\ ker(α on H^{∗}(M_{12}, 𝔽_{2})) & q ≡ 1, 3 (mod 4) \\ ker(x on H^{∗}(M_{12}, 𝔽_{2})/(α)) & q ≡ 2 (mod 4). \end{matrix}

From the presentation for H^{∗}(M_{12}, 𝔽_{2}) given in Theorem 2.1, we see that E_{4}^{∗,0} = H^{∗}(M_{12}, 𝔽_{2})/(α, x) is generated by y, z, β, γ, δ, ε subject to the relations (1)–(8) of Theorem 1.1.

The elements z, β, δ form a system of parameters, and as a module for 𝔽_{2}[z, β, δ], E_{4}^{∗,0} is a direct sum of a free module on three generators 1, γ, ε, and the module 𝔽_{2}[β] · y.

As a module over E_{4}^{∗,0}, E_{4}^{∗,1} is generated by zt and γt subject to the relations

(9′) y · zt = 0
(10′) y · γt = 0
(11′) z · γt = γ · zt
(12′) γ · γt = (ε + zβ) · zt
(13′) ε · γt = zδ · zt.

Thus as a module for 𝔽_{2}[z, β, δ], E_{4}^{∗,1} is free on three generators zt, ε · zt and γt.

Similarly, as a module over E_{4}^{∗,0}, E_{4}^{∗,2} is generated by zt^{2}, γt^{2}, δt^{2} and εt^{2} subject to the relations

(14′) y · zt^{2} = 0  (15′) y · γt^{2} = 0
(16′) y · δt^{2} = 0  (17′) y · εt^{2} = 0
(18′) z · γt^{2} = γ · zt^{2}  (19′) γ · γt^{2} = (ε + zβ) · zt^{2}
(20′) ε · γt^{2} = zδ · zt^{2}  (21′) z · δt^{2} = δ · zt^{2}
(22′) z · εt^{2} = ε · zt^{2}  (23′) γ · εt^{2} = zδ · zt^{2}
(24′) (ε + zβ) · εt^{2} = γδ · zt^{2}  (25′) γ · δt^{2} = δ · γt^{2}
(26′) δ · εt^{2} = ε · δt^{2}

t^{4}  …  …

↑ τ

zt^{3}  γt^{3} z^{2}t^{3} zβt^{3} zγt^{3} zδt^{3} z^{2}βt^{3} zβ^{2}t^{3} zβγt^{3} z^{3}βt^{3} z^{2}β^{2}t^{3}
↑  ↑  z^{3}t^{3} zεt^{3} z^{2}γt^{3} z^{4}t^{3} zβδt^{3} zγδt^{3}
ρ  ω  βγt^{3}  γδt^{3} z^{2}δt^{3} z^{2}εt^{3} zβεt^{3}
β^{2}γt^{3} z^{3}γt^{3}
zt^{2}  γt^{2} z^{2}t^{2} zβt^{2} zγt^{2} zδt^{2} z^{2}βt^{2} zβ^{2}t^{2} zβγt^{2} z^{3}βt^{2} z^{2}β^{2}t^{2}
↑  ↑ δt^{2} εt^{2}  z^{3}t^{2} zεt^{2} z^{2}γt^{2} z^{4}t^{2} zβδt^{2} zγδt^{2}
ξ  η ↑  ↑  βγt^{2} βδt^{2} γδt^{2} z^{2}δt^{2} z^{2}εt^{2} zβεt^{2}
χ ψ  βεt^{2} δ^{2}t^{2} β^{2}γt^{2} z^{3}γt^{2}
δεt^{2} β^{2}δt^{2}
zt  γt z^{2}t zβt zγt zδt z^{2}βt zβ^{2}t zβγt z^{3}βt z^{2}β^{2}t
↑  ↑  z^{3}t zεt z^{2}γt z^{4}t zβδt zγδt
θ  ϕ  βγt  γδt z^{2}δt z^{2}εt zβεt
β^{2}γt z^{3}γt
1  y β γ z^{2} yβ zγ z^{3} z^{2}β yβ^{2} z^{4}  z^{3}β z^{2}β^{2}
z  δ zβ β^{2} zδ  zε zβ^{2} zβγ zβδ zγδ
ε  βγ βδ z^{2}γ z^{2}δ z^{2}ε zβε
βε  β^{3} β^{2}γ z^{3}γ
γδ  δ^{2}  δε  β^{2}δ

E_{4}^{∗,∗} for the Lyndon–Hochschild–Serre spectral sequence

Thus as a module over 𝔽_{2}[z, β, δ], E_{4}^{∗,2} has two free summands generated by γt^{2} and εt^{2}, and another summand generated by zt^{2} and δt^{2} satisfying the relation z · δt^{2} = δ · zt^{2}.

Finally, as a module over E_{4}^{∗,0}, E_{4}^{∗,3} is isomorphic to E_{4}^{∗,1}, the isomorphism being given by multiplication by t^{2}. So it is generated by zt^{3} and γt^{3} subject to the relations

(27′) y · zt^{3} = 0  (28′) y · γt^{3} = 0
(29′) z · γt^{3} = γ · zt^{3}  (30′) γ · γt^{3} = (ε + zβ) · zt^{3}
(31′) ε · γt^{3} = zδ · zt^{3}.

This completes the calculation of E_{4}. Again applying Serre’s transgression theorem, we see that t^{4} survives to E_{5} in order to transgress, and then

d_{5}(t^{4}) = d_{5}(Sq^{2}(t^{2})) = Sq^{2}(d_{3}(t^{2})) = Sq^{2}(x) = αx

which is zero in E_{5}. It follows that t^{4} survives to E_{∞}. So all the differentials commute with multiplication by t^{4}, and each page is periodic in the vertical direction with period four. It only remains to determine whether there is a nonzero differential d_{4} : E_{4}^{∗,3} → E_{4}^{∗,0}. We have therefore proved the following.

Proposition 3.2. There exists an element τ ∈ H^{4}(2M_{12}, 𝔽_{2}) which represents the element t^{4} ∈ E_{∞}^{0,4}. In particular, we have E_{5} = E_{∞}.

4. The differential d_{4}

In order to complete the calculation of the E_{∞} page of the Lyndon–Hochschild– Serre spectral sequence, it suffices to calculate the differential d_{4} : E_{4}^{∗,3} → E_{4}^{∗,0}. Since this is a map of modules over E_{4}^{∗,0}, it suffices to calculate

d_{4}(zt^{3}) ∈ E_{4}^{7,0} = span(yβ, zβ, ε)

and

d_{4}(γt^{3}) ∈ E_{4}^{9,0} = span(z^{3}, zδ, βγ).

We shall prove that d_{4} = 0 using the Poincaré duality method of [2]. This requires us to begin by choosing a homogeneous set of parameters in H^{∗}(2M_{12}, 𝔽_{2}). We choose z, β, δ and a representative τ ∈ H^{4}(2M_{12}, 𝔽_{2}) of t^{4} ∈ E_{∞}^{0,4}. We thus obtain a spectral sequence which we write using a bold typeface to distinguish it from the Lyndon– Hochschild–Serre spectral sequence,

E_{2}^{∗,∗} = H^{∗}(2M_{12}, 𝔽_{2}) ⊗ Λ(z̃, β̃, δ̃, τ̃)

with H^{∗}(2M_{12}, 𝔽_{2}) along the base, z̃, β̃, δ̃ and τ̃ in degrees 2, 3, 5 and 3 respectively up the fibre, and converging to finite dimensional graded vector space satisfying Poincaré duality with top class in degree 2 + 3 + 5 + 3 = 13 on the base. Moreover, we have d_{3}(z̃) = z, d_{4}(β̃) = β, d_{6}(δ̃) = δ and d_{4}(τ̃) = τ, so that in E_{∞}, the base is some quotient of H^{∗}(2M_{12}, 𝔽_{2})/(z, β, δ, τ). Now the latter is already zero in degree bigger than 13, and in degree 13 it is spanned by a representative of zεt^{3} provided this survives to E_{∞}, and is zero otherwise. In particular, the existence of a non-zero class in E_{∞}^{13,0} forces zεt^{3} to survive to E_{∞}^{10,3}. Thus d_{4}(zt^{3}) is in the kernel of multiplication by ε, so it is either zero or yβ. The relation γ.d_{4}(zt^{3}) = z.d_{4}(γt^{3}) now forces d_{4}(γt^{3}) = 0.

To determine whether d_{4}(zt^{3}) is equal to zero or to yβ, we use a theorem of Duflot [10], which states that the depth of the mod p cohomology ring of a finite group is at least equal to the p-rank of the center of the Sylow p-subgroup. For 2M_{12} in characteristic two, this means that the depth has to be at least two. If d_{4}(zt^{3}) were equal to yβ, then y ∈ H^{∗}(2M_{12}, 𝔽_{2}), would annihilate z, β and δ, and so the cohomology ring modulo the annihilator of y would have Krull dimension one. This cannot happen in a ring of depth two. Hence we deduce that d_{4}(zt^{3}) = 0.

At this stage, we still have several possibilities for what is happening in E_{∞}^{3,7}, E_{∞}^{5,5} and E_{∞}^{11,0}. Poincaré duality forces the element γϕ = (ε + zβ)θ ∈ E_{2}^{11,0} to be hit either by ξδ̃ or by (ξ + γ)δ̃ in E_{2}^{3,7}, or by yz̃δ̃ ∈ E_{2}^{5,5}. Later, we shall see that in fact (ξ + γ)δ gives a nonzero element in E_{∞}^{5,5}, while d_{8}(yz̃δ̃) = γψ. This corresponds to a nontrivial Massey product ⟨z, y, δ⟩ = γϕ in H^{∗}(2M_{12}, 𝔽_{2}) (well defined modulo the ideal generated by z and δ). See Section 11 of [2] for an explanation of the relevance of the Massey products here, and note that the Massey product ⟨z, y, δ⟩ is the same as the element ⟨y|z, δ⟩ defined there, because we are working with trivial coefficients. Although z, y and δ are all inflations from H^{∗}(M_{12}, 𝔽_{2}), the Massey product is not. This can happen because the relations allowing the definition of the Massey product do not hold in the cohomology of M_{12}.

[yz̃δ̃]

[δ̃]  yδ̃ (ξ + γ)δ̃

[β̃], [τ̃]

[z̃]  yz̃

1  y  θ  ξ ρ η χ ψ [γϕ = (ε + zβ)θ] ερ
γ ϕ ε ω

E_{∞}^{∗∗} for the Poincaré duality spectral sequence (elements in brackets have disappeared by the E_{∞} page)

We now have enough information to verify the Poincaré series for H^{∗}(2M_{12}, 𝔽_{2}) given in Theorem 1.1.

5. Ungrading the E_{∞} page

We use the same names for elements of H^{∗}(M_{12}, 𝔽_{2}) and their inflations to elements of H^{∗}(2M_{12}, 𝔽_{2}), which gives us the generators y, z, β, γ, δ, ε and relations (1)–(8) of Theorem 1.1. Our tasks are to choose representatives in H^{∗}(2M_{12}, 𝔽_{2}) for zt, γt, zt^{2}, γt^{2}, δt^{2}, εt^{2}, zt^{3}, γt^{3} and t^{4} in E_{∞}^{∗∗}, to ungrade the module structure over the image of inflation, and to evaluate the products of these representatives.

Write Ê_{a}, Ê_{b} and Ê_{c} for the inverse images in 2M_{12} of the elementary abelian subgroups E_{a}, E_{b} and E_{c} of M_{12}. By looking at the restrictions of the extension class α to these elementary abelian subgroups, we see that Ê_{a} ≅ (ℤ/2)^{4}, Ê_{b} ≅ ℤ/4×(ℤ/2)^{2} and Ê_{c} ≅ Q_{8} × ℤ/2.

Since E_{∞}^{9,0} = γ.E_{∞}^{4,0} ⊕ z.E_{∞}^{6,0}, we see that there exist unique representatives θ ∈ H^{4}(2M_{12}, 𝔽_{2}) of zt ∈ E_{∞}^{3,1} and ϕ ∈ H^{6}(2M_{12}, 𝔽_{2}) of γt ∈ E_{∞}^{5,1} satisfying γθ + zϕ = 0. Since zδθ + εϕ is an element of E_{∞}^{9,0} killed by multiplication by γ, it must be zero. Similarly, (ε + zβ)θ + γϕ is an element of E_{∞}^{11,0} killed by multiplication by γ, so it is either zero or yβ^{2}. But it restricts to zero on Ê_{c} (because ε and γ do) whereas yβ^{2} doesn’t, so in fact (ε + zβ)θ + γϕ = 0. This proves relations (11), (12) and (13).

Next, we observe that multiplication by z induces an injection from H^{5}(2M_{12}, 𝔽_{2}) to H^{8}(2M_{12}, 𝔽_{2}), and the cokernel is spanned by β^{2}. So we may choose a representative ξ ∈ H^{5}(2M_{12}, 𝔽_{2}) of zt^{2} ∈ E_{∞}^{3,2} in such a way that θ^{2} + zξ is either zero or β^{2}. Then γ(θ^{2} + zξ) = z(ϕθ + γξ) is either zero or β^{2}γ. It is easy to check that β^{2}γ cannot be a multiple of z, and so this forces θ^{2} = zξ. Then z(ϕθ + γξ) = 0 and so ϕθ = γξ. Similarly, z(ε + zβ)ξ = γ^{2}ξ = γϕθ = zϕ^{2} and so ϕ^{2} = (ε + zβ)ξ. This proves relations (32), (33) and (34).

We choose a representative η ∈ H^{7}(2M_{12}, 𝔽_{2}) of γt^{2} ∈ E_{∞}^{5,2} in such a way that γξ+zη is some linear combination of βϕ, δθ and βδ. Then γ(γξ + zη) = z(ε + zβ)ξ + zγη is some linear combination of βγϕ = β(ε + zβ)θ, γδθ and βγδ. Divisibility by z then forces this linear combination to be zero, and so we have γξ = zη. Since zδξ + εη and (ε + zβ)ξ + γη are both killed by multiplication by γ, they must be zero. This proves relations (18), (19) and (20).

We choose a representative ψ ∈ H^{9}(2M_{12}, 𝔽_{2}) of εt^{2} in such a way that zψ + εξ is some linear combination of β^{2}θ, δϕ, β^{3} and δ^{2}. Then

(ε + zβ)(zψ + εξ) = z((ε + zβ)ψ + γδξ) = γ(γψ + zδξ)

is some linear combination of β^{2}(ε+zβ)θ = β^{2}γϕ, δ(ε+zβ)ϕ = zδ(δθ+βϕ), β^{3}(ε+zβ) and δ^{2}(ε + zβ). Divisibility by both z and γ forces this linear combination to be zero and so we have zψ = εξ. Then γ^{2}ψ = z(ε+zβ)ψ = ε(ε+zβ)ξ = zγδξ and so γψ = zδξ and (ε + zβ)ψ = γδξ. This proves relations (22), (23) and (24).

Since yθ is killed by multiplication by z, it is either zero or equal to yβ. But if yθ = yβ then 0 = yzξ = yθ^{2} = yθβ = yβ^{2} ≠ 0. So we must have yθ = 0. Since yϕ, yξ, yη, yψ and yρ are killed by multiplication by z, they are zero. This proves relations (9), (10), (14), (15), (17) and (27).

6. Cohomology of the Sylow 2-subgroup

At this stage, it is hard to get further in ungrading the relations on the E_{∞} page of the Lyndon–Hochschild–Serre spectral sequence by theoretical means alone. So we resort to calculation using the MAGMA computer algebra package. This involves constructing explicit projective resolutions of modules and chain maps. Products are computed directly from the Yoneda definition, via composition of chain maps. These programs are currently only effective in the case of p-groups, so it is not possible to compute the cohomology of 2M_{12} directly this way, so we must work with a Sylow 2-subgroup. See the Appendix for a power-commutator presentation of the Sylow 2-subgroup of 2M_{12}.

We use the letters z, y, x, … to denote the generators of the cohomology of each group in question. This means that there is some ambiguity in the notation, and so we need to specify the ambient cohomology ring every time we use one of these symbols. In order to minimize the possibility of errors creeping in, we have made no attempt to rechoose the generators in such a way that the relations simplify, or in such a way that restriction and inflation maps simplify. However, the programs do return elements of these rings in normal form with respect to suitable Gröbner bases, and when we give restriction and inflation maps, the answers are consistently given with respect to these bases.

Theorem 6.1. The cohomology of the Sylow 2-subgroup P of M_{12} is the quotient of the polynomial ring generated by elements z, y, x, w, v, u, t, of degrees 1, 1, 1, 2, 2, 3, 4, by the relations

(i) zy, (ii) zx, (iii) zw, (iv) yv, (v) yxw + x^{2}w + w^{2}, (vi) wv + xu, (vii) yu, (viii) x^{2}u + wu, (ix) zvu + xvu + z^{2}t + u^{2}.

There are five conjugacy classes of maximal elementary abelian 2-subgroups, and each has rank three and cohomology 𝔽_{2}[z, y, x]. The restrictions of the above generators of the cohomology of P are as follows.

E_{1}: z ↦ x, y ↦ 0, x ↦ 0, w ↦ 0, v ↦ y^{2} + yx, u ↦ z^{2}x + zx^{2}, t ↦ z^{4} + z^{2}y^{2} + z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2}. The kernel of restriction is generated by y, x, w.

E_{2}: z ↦ 0, y ↦ 0, x ↦ x, w ↦ 0, v ↦ y^{2} + yx, u ↦ 0, t ↦ z^{4} + z^{2}y^{2} + z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2}. The kernel of restriction is generated by z, y, w, u.

E_{3}: z ↦ 0, y ↦ 0, x ↦ z, w ↦ z^{2}, v ↦ zy + y^{2}, u ↦ z^{2}y + zy^{2}, t ↦ z^{4} + z^{3}y + z^{2}y^{2} + z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2} + y^{2}x^{2} + x^{4}. The kernel of restriction is generated by z, y, x^{2} + w, xv + u.

E_{4}: z ↦ 0, y ↦ z, x ↦ y, w ↦ 0, v ↦ 0, u ↦ 0, t ↦ z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2} + y^{2}x^{2} + x^{4}. The kernel of restriction is generated by z, w, v, u.

E_{5}: z ↦ 0, y ↦ z + y, x ↦ y, w ↦ zy, v ↦ 0, u ↦ 0, t ↦ zy^{3} + z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2} + y^{2}x^{2} + x^{4}. The kernel of restriction is generated by z, yx + x^{2} + w, v, u.

The restriction map from H^{∗}(M_{12}, 𝔽_{2}) to H^{∗}(P, 𝔽_{2}) is calculated using the fact that the latter ring has no nilpotent elements, or equivalently, that elements can be distinguished by their restrictions to elementary abelian 2-subgroups. This requires us simultaneously to calculate the correspondence between the above names for the generators of the cohomology of the elementary abelian 2-subgroups and the names used in Section 2. There is some ambiguity in this correspondence, given by the action of the normalizer in M_{12}, and up to this ambiguity, it is easy to calculate that the following holds.

Proposition 6.2. The elementary abelian subgroups E_{1} and E_{2} of P are conjugate to E_{a} in M_{12}, E_{3} and E_{4} in P are conjugate to E_{b} in M_{12}, and E_{5} in P is conjugate to E_{c} in M_{12}. Conjugation acts on cohomology as follows:

\begin{matrix}E_{a} & E_{1} & E_{2} \\ u & x & x \\ v & y + z & y \\ w & z & z \end{matrix}  \begin{matrix}E_{b} & E_{3} & E_{4} \\ u & z & z \\ v & y & y \\ w & x & x \end{matrix}  \begin{matrix}E_{c} & E_{5} \\ u & z \\ v & y \\ w & x \end{matrix}

Proposition 6.3. The restriction map from H^{∗}(M_{12}, 𝔽_{2}) to H^{∗}(P, 𝔽_{2}) is given as follows: res(α) = y^{2} + w (degree 2) res(x) = yw (degree 3) res(y) = y^{3} + xw (degree 3) res(z) = zv + xv + u (degree 3) res(β) = z^{4} + y^{2}x^{2} + x^{4} + x^{2}w + w^{2} + v^{2} + xu + t (degree 4) res(γ) = z^{3}v + x^{3}v + zv^{2} + xv^{2} + z^{2}u + wu + vu + zt (degree 5) res(δ) = xwu + z^{2}t + yxt + x^{2}t + u^{2} + wt + vt (degree 6) res(ε) = x^{3}w^{2} + xw^{3} + y^{2}xt + yx^{2}t + w^{2}u + xu^{2} + zvt + xvt (degree 7)

Next, we calculate the cohomology of the Sylow 2-subgroup of 2M_{12}, and the inflation map from the Sylow 2-subgroup of M_{12}.

Theorem 6.4. The cohomology of the Sylow 2-subgroup P̂ of 2M_{12} is the quotient of the polynomial ring generated by elements z, y, x, w, v, u, t, s, r, q, p, n, m, k, j, of degrees 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 6, by the relations

(i) zy, (ii) zx, (iii) y^{3}, (iv) y^{2}x + yv, (v) yw + yv, (vi) xw + yv, (vii) w^{2} + zt, (viii) yu, (ix) xu, (x) yt, (xi) xt, (xii) z^{3}w + wu + zr + zq, (xiii) zwv + wu + vu + wt + zq + zp, (xiv) z^{2}t + wu + vu + zq, (xv) zs + zr + zq, (xvi) ys, (xvii) xs + yr, (xviii) yq, (xix) yp, (xx) zwu + zvu + z^{2}q + ut + wr + wq, (xxi) zwt + zvt + t^{2} + wp, (xxii) z^{2}r + z^{2}q + vs + wq + wp + zk, (xxiii) x^{2}q + vs + vr + vq, (xxiv) z^{2}p + vs + wr, (xxv) z^{2}n + vs + wq + zk, (xxvi) y^{2}n + yk, (xxvii) z^{2}m + u^{2}, (xxviii) ws + wr + wq, (xxix) z^{2}vu + zwm + z^{2}k + ur + tr + uq + tq + tp + zj, (xxx) yx^{2}r + yj, (xxxi) z^{3}q + zwn + zwm + zvm + z^{2}k + tr + uq + tq + zj, (xxxii) zut + zwq + zwp + zwn + zwm + zvm + yxk + uq + up + wk, (xxxiii) zwq + zwp + zwn + yxk + up + wk, (xxxiv) zwr + zwn + tr + tq + up + tp, (xxxv) zwp + zwn + yxk + tr + tq + wk, (xxxvi) zvp + zwn + zvn + tq + wk + vk, (xxxvii) us + ur + uq, (xxxviii) ts + tr + tq, (xxxix) x^{4}r + x^{3}k + x^{2}j + sp + rp + qp, (xl) yx^{3}m + s^{2} + sr + sq, (xli) x^{4}m + s^{2} + r^{2} + q^{2}, (xlii) wut + vut + ztq + ztp + wvm + v^{2}m + zwk + sq + q^{2} + tk + wj, (xliii) x^{2}vr + z^{2}vq + wvq + ztq + wvn + zun + wvm + v^{2}m + zwk + zvk + s^{2} + sq + q^{2} + sp + qp + uk + vj, (xliv) x^{2}vm + sq + rq + q^{2}, (xlv) wu^{2} + vu^{2} + zuq + zup + zun + ztm + uk, (xlvi) vut + ztp + zun + ztn + v^{2}m + ztm + s^{2} + q^{2} + sp + uk + wj, (xlvii) ztp + wvm + ztm + zwk + sq + tk + wj, (xlviii) v^{2}s+v^{2}r+v^{2}q+ztq+wvp+zup+zun+wvm+ztm+s^{2}+sq+sp+p^{2}+uk+wj, (xlix) wvr + wvq + ztq + zup + zun + wvm + ztm + s^{2} + sq + uk + tk + wj,

(l) zur + zuq + ztq + zun + ztn + wvm + ztm + zwk + sq + wj, (li) ztr + zup + zun + ztn + zwk + s^{2} + sp + uk + tk, (lii) z^{2}uq + zr^{2} + zq^{2} + wtp + zrp + zqp + zp^{2} + vun + zqn + zpn + wum + vum + zqm + zuk + sk + pk + uj, (liii) zr^{2} + zq^{2} + wtp + zqp + zp^{2} + zrn + zpn + wtm + vtm + wvk + qk + tj, (liv) ut^{2} + vup + wtp + zqp + wun + vun + zqn + wvk + ztk + pk + tj, (lv) wtp + zqp + zqn + wtm + vtm + wvk + qk + pk + tj, (lvi) vuq+zq^{2}+vup+zrp+zqp+zrn+wtm+vtm+zpm+wvk+ztk+sk+qk+pk+tj, (lvii) yrn + sk + rk + qk, (lviii) utq + utp + utn + wpm + wuk + vuk + zqk + zpk + znk + ynk + k^{2}, (lix) yxr^{2} + vsq + wrq + wrp + wp^{2} + utn + wpn + t^{2}m + wpm + wuk + vuk + wtk + zpk + ynk + ztj + k^{2} + sj, (lx) x^{2}r^{2} + x^{2}pm + vr^{2} + vrq + wq^{2} + wrp + wp^{2} + utn + vsn + wqn + t^{2}m + wpm + wuk + vuk + wtk + zrk + xrk + zqk + zpk + znk + ynk + ztj + k^{2} + rj + qj, (lxi) z^{2}q^{2} + vr^{2} + vsq + wrq + vrq + utn + vsn + wpn + vpm + wuk + vuk + zrk + zqk + zpk + ynk + k^{2} + qj, (lxii) x^{2}rp + wrq + vrq + vq^{2} + utp + wp^{2} + zrk + zqk + zpk + znk + pj, (lxiii) z^{2}qk+ur^{2}+uq^{2}+up^{2}+urn+uqn+upn+utk+wqk+wnk+zk^{2}+xk^{2}+ynj+kj, (lxiv) wvuk + zwmk + zvmk + x^{2}rj + z^{2}qj + sq^{2} + rq^{2} + q^{3} + srp + r^{2}p + rqp + urk + trk + unk + tnk + wk^{2} + utj + wpj + zkj + xkj + j^{2}. The Poincaré series of this ring is

∑_{i=1}^{∞} t^{i} dim_{𝔽_{2}} H^{i}(P̂, 𝔽_{2}) = (1 + t + t^{2} + 2t^{4} + t^{5})/(1 − t)^{2}(1 − t^{4})^{2}

= 1 + 3t + 6t^{2} + 9t^{3} + 16t^{4} + 26t^{5} + 38t^{6} + 50t^{7} + 69t^{8} + 93t^{9}

+ 120t^{10} + 147t^{11} + 184t^{12} + 228t^{13} + 276t^{14} + 324t^{15} + …

Inflation from H^{∗}(P, 𝔽_{2}) to H^{∗}(P̂, 𝔽_{2}) is given by inf(z) = z (degree 1) inf(y) = y (degree 1) inf(x) = x (degree 1) inf(w) = y^{2} (degree 2) inf(v) = y^{2} + yx + v (degree 2) inf(u) = zw + u + t (degree 3) inf(t) = z^{2}w + w^{2} + s + r + p + n + m (degree 4). Denote by Ê_{1}, …, Ê_{5} the inverse images in P̂ of the elementary abelian subgroups E_{1}, …, E_{5} of P. (See the Appendix for these subgroups in terms of the presentation for P̂.) Then we have the following. Ê_{1} ≅ (ℤ/2)^{4}, H^{∗}(Ê_{1}, 𝔽_{2}) = 𝔽_{2}[z, y, x, w], where z, y, x and w have degree one. The inflation map from E_{1} to Ê_{1} is given by inf(z) = z, inf(y) = y, inf(x) = x. Ê_{2} ≅ (ℤ/2)^{4}, H^{∗}(Ê_{2}, 𝔽_{2}) = 𝔽_{2}[z, y, x, w], where z, y, x and w have degree one. The inflation map from E_{2} to Ê_{2} is given by inf(z) = z, inf(y) = y, inf(x) = x. Ê_{3} ≅ ℤ/4 × (ℤ/2)^{2}, H^{∗}(Ê_{3}, 𝔽_{2}) = 𝔽_{2}[z, y, x, w]/(z^{2}), where z, y and x have degree one and w has degree two. The inflation map from E_{3} to Ê_{3} is given by inf(z) = z, inf(y) = y, inf(x) = x. Ê_{4} ≅ ℤ/4 × (ℤ/2)^{2}, H^{∗}(Ê_{4}, 𝔽_{2}) = 𝔽_{2}[z, y, x, w]/(z^{2}), where z, y and x have degree one and w has degree two. The inflation map from E_{4} to Ê_{4} is given by inf(z) = z, inf(y) = y, inf(x) = x. Ê_{5} ≅ Q_{8} × ℤ/2, H^{∗}(Ê_{5}, 𝔽_{2}) = 𝔽_{2}[z, y, x, w]/(z^{2} + zy + y^{2}, y^{3}), where z, y and x have degree one and w has degree four. The inflation map from E_{5} to Ê_{5} is given by inf(z) = z, inf(y) = y, inf(x) = x.

By restricting from M_{12} to P and then inflating to P̂, we obtain the following restrictions from H^{∗}(2M_{12}, 𝔽_{2}) to H^{∗}(P̂, 𝔽_{2}):

res(y) = yv
res(z) = yx^{2} + zw + zv + yv + xv + u + t
res(β) = z^{4} + x^{4} + z^{2}w + v^{2} + zt + s + r + p + n + m
res(γ) = yx^{4} + z^{3}v + x^{3}v + zv^{2} + xv^{2} + z^{2}u + wu + wt + vt + zn + zm
res(δ) = zvu + yxr + x^{2}r + x^{2}p + x^{2}n + x^{2}m + vs + vr + vp + vn + vm
res(ε) = zt^{2} + zvr + xvr + xvp + xvn + zwm + xvm + yxk + uq + tq + up + vk.

The next task is to identify in H^{∗}(P̂, 𝔽_{2}) the restrictions of the elements denoted θ, ϕ, etc., in Section 5. The first step in this process is to identify elements of H^{∗}(P̂, 𝔽_{2}) with the property that the restrictions to Ê_{1} and Ê_{2} are conjugate under the inflation of the correspondence described in Proposition 6.2 and invariant under the action of the normalizer in 2M_{12}, and similarly for Ê_{3} and Ê_{4}, and for Ê_{5}. This was carried out in degrees ≤ 9. Among the elements of this subring, it turns out that the following hold:

1) There exist unique nonzero elements res_{2M_{12},P̂}(θ) and res_{2M_{12},P̂}(ϕ) such that relation (11) of Theorem 1.1 (zϕ = γθ) holds with the restrictions of z and γ given above, i.e.,

res_{2M_{12},P̂}(z)res_{2M_{12},P̂}(ϕ) = res_{2M_{12},P̂}(γ)res_{2M_{12},P̂}(θ).

Namely,

res_{2M_{12},P̂}(θ) = z^{2}w + s + p,
res_{2M_{12},P̂}(ϕ) = zvu + z^{2}q + x^{2}p + yxm + ut + vs + wr + wp + vp + wn + wm + xk.

2) There exists a unique element res_{2M_{12},P̂}(ξ) such that relation (32) of Theorem 1.1 (θ^{2} = zξ) holds, namely

res_{2M_{12},P̂}(ξ) = wu + vu + zr + yr + xq + zp + yn + k.

3) There exists a unique element res_{2M_{12},P̂}(η) such that relation (18) of Theorem 1.1 (zη = γξ) holds, namely

res_{2M_{12},P̂}(η) = yx^{2}m + zvr + xvr + zvq + zwn + zvn

+ zwm + zvm + yxk + x^{2}k + uq + tp + tn + tm + wk.

4) There exists a unique element res_{2M_{12},P̂}(ψ) such that relation (23) of Theorem 1.1 (γψ = zδξ) holds, namely

res_{2M_{12},P̂}(ψ) = x^{4}k + yx^{2}j + t^{3} + yr^{2} + vtq + xrq

+ vup + vtp + zqp + xqp + vtn + xqn + vtm + xqm + wvk + pk + tj.

5) There exists a unique element res_{2M_{12},P̂}(χ) such that the restriction of zχ + δξ is a linear combination of εθ, yβ^{2}, βε and γδ, namely

res_{2M_{12},P̂}(χ) = x^{4}p + x^{4}n + z^{3}k + wvq + v^{2}q + v^{2}p + zup + wvn

+ v^{2}n + ztm + zvk + z^{2}j + x^{2}j + s^{2} + qp + p^{2} + rn + rm + uk.

With this information, it now follows that zχ + δξ is equal to γδ, so that relation (21) of Theorem 1.1 holds.

6) There exists a unique element res_{2M_{12},P̂}(ρ) such that the restriction of θξ + zρ is a linear combination of the restrictions of βξ, ψ and βγ, namely

res_{2M_{12},P̂}(ρ) = zvt + yxr + x^{2}r + z^{2}q + ut + t^{2} + wr + wq + wp + zk + j.

With this information, it now follows that θξ + zρ is equal to zero, so that relation (35) of Theorem 1.1 holds.

7) There exists a unique element res_{2M_{12},P̂}(ω) such that the restriction of zω + γρ is a linear combination of βη, εθ and yβ^{2}, namely

res_{2M_{12},P̂}(ω) = x^{3}k + wt^{2} + wvq + ztq + wvn + ztm + yxj + p^{2} + pn + pm + tk.

8) There exists a unique element res_{2M_{12},P̂}(τ) such that the restriction of ξ^{2} + z^{2}τ is a linear combination of the restrictions of βρ, γξ, δθ, βϕ, zε and βδ, namely

res_{2M_{12},P̂}(τ) = z^{2}w + s + r + p + n.

With this information, it now follows that ξ^{2} +z^{2}τ is equal to zε, so that relation (43) of Theorem 1.1 holds.

This completes the determination of the restriction map from H^{∗}(2M_{12}, 𝔽_{2}) to H^{∗}(P̂, 𝔽_{2}). Now in order to ungrade the remaining relations on the E_{∞} page of the Lyndon–Hochschild–Serre spectral sequence for H^{∗}(2M_{12}, 𝔽_{2}), in each case the associative law for multiplication determines the relation up to adding elements of the form yβ^{n}. Restriction to P̂ now determines the ungraded relations completely, and in fact it turns out that yβ^{n} never features in any of the relations. This means that the ring extension of H^{∗}(2M_{12}, 𝔽_{2})/⟨y⟩ by ⟨y⟩ is trivial.

The action of the Steenrod operations were calculated by restricting to the subgroups Ê_{1}, Ê_{4} and Ê_{5} of P̂, which are conjugate to Ê_{a}, Ê_{b} and Ê_{c} in 2M_{12}. In each case, there is a unique element of H^{∗}(2M_{12}, 𝔽_{2}) with the appropriate restrictions.

7. Notes on Computer Calculations

As has been noted several times a huge amount of the information about the cohomology of 2M_{12} was computer generated. In addition the computer was used extensively to check and verify the calculations that had been done by hand. All of the calculations were performed using the computer algebra package MAGMA [3] on an SUN ULTRA 2200.

The calculations used the programs developed by the second author for computing the cohomology of p-groups. Indeed, we should emphasize that the methods work directly only for p-groups, though with the fusion techniques outlined below some information about the cohomology of more general groups can be gathered. The method for performing the routine operations such as taking projective resolutions, lifting cocycles to chain maps, and computing products and relations among the cohomology generators have all been described in other papers [5, 6]. A description of the algorithm for computing the restriction maps is given in [7]. The method used for getting inflations is very similar. That is, the main operation is that of computing a chain map from the projective resolution of the trivial module for the group to the projective resolution of the trivial module for the quotient group. Then the image of an element of cohomology for the quotient group is the lift of a representing cocycle to a cocycle for the group. Some further explanation of the methods is given on the second author’s web page (http://www.math.uga.edu/∼jfc/groups2.html).

The data on the cohomology rings of the Sylow 2-subgroups of M_{12} and 2M_{12} and their restrictions and inflation in Section 6 were obtained using these programs. Of course most of the results for the Sylow 2-subgroup of M_{12} were already available from [1]. In this case the completeness of the result could be verified by referring to that paper. For the cohomology ring of the Sylow 2-subgroup of 2M_{12} the completeness of the calculation can be verified using the theorem of [4] or by comparing it with the spectral sequence. That is, in the notation of the last section with H the kernel of the quotient map P̂ → P, the Lyndon–Hochschild–Serre spectral sequence

H^{r}(P, H^{s}(H, 𝔽_{2})) ⇒ H^{r+s}(P̂, 𝔽_{2})

must collapse at the E_{5} page because there is an element of degree 4 that restricts nontrivially to the cohomology of H.

Once the cohomology of the Sylow 2-subgroups had been computed, we turned to the task of getting a computer version of the cohomolgy H^{∗}(2M_{12}, 𝔽_{2}). For this a fusion program was written. The basic algorithm underlying the program can be described as follows. Suppose we are given subgroups H and K of P̂ and an element g ∈ 2M_{12} such that H^{g} ⊆ K. Let c_{g} : H → K be the map induced by conjugation by g. Then for each degree n within a reasonable range we form the linear transformations of the vector spaces as in the diagram

\xymatrix{H^{n}(P̂, 𝔽_{2}) \ar[r]^{^{res^{P̂,K}}} \ar[dr]^{_{res_{P̂,H}}} & H^{n}(K, 𝔽_{2}) \ar[d]^{res_{K,H^{g}} ∘c_{g}} \\  & H^{n}(H, 𝔽_{2})}

The fusion space is going to be the null space of the linear transformation

res_{K,H^{g}} ∘ c_{g} ∘ res_{P̂,K} − res_{P̂,H},

namely the space on which the diagram commutes. As noted in Theorem 6.4, the cohomology ring H^{∗}(P̂, 𝔽_{2}) has the form R/I where R is a polynomial ring and I is the ideal of relations among the generators of R. Then a Gröbner basis B for I is a set of elements in I with the property that for any element f ∈ I, the leading term of f is divisible by the leading of some element of B. The notion of “leading term” is based on some prechosen ordering of the monomials of P. The space H^{n}(P̂, 𝔽_{2}) has a basis b consisting of the collection of all monomials of degree n in R that are not leading terms of any element of B. Then the matrix for the map res_{P̂,K} is constructed by taking the restriction of each element of b and expressing it as a linear combination of the basis of monomials that span H^{n}(K, 𝔽_{2}). This last is a “normal form” operation, the normal form being taken with respect to the Gröbner basis of the ideal of relations of H^{∗}(K, 𝔽_{2}).

In fact, the cohomology of 2M_{12} (as a subalgebra of H^{n}(P̂, 𝔽_{2})) is the intersection of several fusion space. the fusions that we had to perform were the following. The subgroups Ê_{1}, …, Ê_{5} were introduced in Section 6.

(i) The conjugation of Ê_{1} into Ê_{2}.

(ii) The conjugation of Ê_{3} into Ê_{4}.

(iii) The automorphism induced on Ê_{2} by an element of order 3 in 2M_{12}.

(iv) The automorphism induced on Ê_{3} by an element of order 3 in 2M_{12}.

(v) The automorphism induced on Ê_{5} by an element of order 3 in 2M_{12}.

The program was designed to identify new generators in each degree. So in the end we had a minimal set of generators for a subalgebra of H^{∗}(P̂, 𝔽_{2}) that was guaranteed to contain the restriction to P̂ of the cohomology ring of 2M_{12}. From the spectral sequence it could be verified that the two rings had the same Poincaré series. Hence they are the same. For the purposes of the rest of this discussion we call the subring A.

The next task was to identify the generators in H^{∗}(2M_{12}, 𝔽_{2}) as represented in the spectral sequence with specific elements of A. We began by identifying the generators of H^{∗}(M_{12}, 𝔽_{2}), with elements in the computed cohomology ring H^{∗}(P, 𝔽_{2}) and taking their inflations. Thus we had the identification for the elements z, y, β, γ, δ and ε identified as elements of A. The other nine generators were identified in an inductive manner by the relations they satisfied with respect to the inflated generators from the cohomology of M_{12}. The existence and uniqueness statements at the end of the previous section were established by a trial and error method from a computer aided investigation of the ring A. Finally we established the 67 relations given in the statement of Theorem 1.1. Most of the relations were predictable up to only a few ambiguities from the spectral sequence. The process was simply a matter of exploring the possibilities and checking to see if the candidate relation held for the identified generators in A. So the relations on the generators of A were computed by hand with a computer verification rather than being computer generated. However, once the relations had been computed then it was easy enough to get a Gröbner basis for the ideal they generate.

Parts of the process were aided by an inverse image program that we wrote to identify the elements of A as polynomials in the given generators. The program was particularly effective in computing the action of the Steenrod algebra on H^{∗}(2M_{12}, 𝔽_{2}). That is, we had established that the cohomology of 2M_{12} is detected on restriction to the subgroups Ê_{1}, Ê_{4} and Ê_{5}. The actions of the Steenrod algebra on the cohomologies of these groups are well known. Consequently for each generator x of A we could take its restrictions to the subgroups Ê_{1}, Ê_{4} and Ê_{5}, and compute the action of the Steenrod algebra on the restrictions, thereby obtaining res_{2M_{12},Ê_{j}}(Sq^{i}(x)) for all relevant i and j. Then we calculate the unique inverse images in A. The algorithm behind the program involved writing the restriction maps res_{2M_{12},Ê_{j}} as linear transformations of the space of homogeneous element of degree n for the appropriate n. It was similar to the program written for the fusions in this regard. In this case we needed the fact that we had computed a Gröbner basis for the ideal of relations among the generators of A.

Appendix: Restriction maps

Here is a power-commutator presentation for P̂ ∈ Syl_{2}(2M_{12}):

Generators g_{1}, …, g_{7}

Relations g_{1}^{2} = 1, g_{2}^{2} = 1, g_{3}^{2} = g_{6}, g_{4}^{2} = g_{6}, g_{5}^{2} = 1, g_{6}^{2} = 1, g_{7}^{2} = 1, [g_{1}, g_{2}] = g_{5}g_{6}g_{7}, [g_{1}, g_{3}] = g_{4}, [g_{1}, g_{4}] = g_{6}, [g_{1}, g_{5}] = 1, [g_{1}, g_{6}] = 1, [g_{2}, g_{3}] = 1, [g_{2}, g_{4}] = g_{6}g_{7}, [g_{2}, g_{5}] = 1, [g_{2}, g_{6}] = 1, [g_{2}, g_{7}] = 1, [g_{3}, g_{4}] = g_{6}, [g_{3}, g_{5}] = g_{6}g_{7}, [g_{3}, g_{6}] = 1, [g_{3}, g_{7}] = 1, [g_{4}, g_{5}] = 1, [g_{4}, g_{6}] = 1, [g_{4}, g_{7}] = 1, [g_{5}, g_{6}] = 1, [g_{5}, g_{7}] = 1, [g_{6}, g_{7}] = 1. (Here, [a, b] denotes aba^{−1}b^{−1}).

We have Z(2M_{12}) = ⟨g_{7}⟩, Z(P̂) = ⟨g_{6}, g_{7}⟩, and Φ(P̂) = [P̂, P̂] = ⟨g_{4}, g_{5}, g_{6}, g_{7}⟩.

The subgroups Ê_{i}, 1 ≤ i ≤ 5 are the inverse images in P̂ of a set of representatives E_{i} for the conjugacy classes of maximal elementary abelian subgroups of P ∈ Syl_{2}(M_{12}). They are given by

Ê_{1} = ⟨g_{1}, g_{5}, g_{6}, g_{7}⟩ ≅ (ℤ/2)^{4},
Ê_{2} = ⟨g_{2}, g_{5}, g_{6}, g_{7}⟩ ≅ (ℤ/2)^{4},
Ê_{3} = ⟨g_{2}g_{4}, g_{5}, g_{6}, g_{7}⟩ ≅ ℤ/4 × (ℤ/2)^{2},
Ê_{4} = ⟨g_{2}g_{5}, g_{3}g_{4}g_{5}, g_{6}, g_{7}⟩ ≅ ℤ/4 × (ℤ/2)^{2},
Ê_{5} = ⟨g_{2}g_{4}, g_{3}g_{4}g_{5}, g_{6}, g_{7}⟩ ≅ Q_{8} × ℤ/2.

We give the restriction maps from P̂ and 2M_{12} to the subgroups Ê_{i}. These subgroups detect cohomology of P̂, and hence also of 2M_{12}. In 2M_{12}, Ê_{1} is conjugate to Ê_{2} and Ê_{3} is conjugate to Ê_{4}, so we only give the restriction maps from 2M_{12} to Ê_{1}, Ê_{3} and Ê_{5}.

Restriction from P̂ to Ê_{1}: z ↦ z + y y ↦ 0 x ↦ 0 w ↦ zx + yx v ↦ zy + zx + yx + x^{2} u ↦ z^{2}x + y^{2}x + zx^{2} + yx^{2} + z^{2}w + y^{2}w + zw^{2} + yw^{2} t ↦ zx^{2} + yx^{2} s ↦ z^{3}x + z^{2}yx + zy^{2}x + y^{3}x + z^{2}x^{2} + y^{2}x^{2} + zx^{3} + yx^{3} + z^{2}xw + y^{2}xw + zxw^{2} + yxw^{2} r ↦ z^{3}x + y^{3}x + zyx^{2} + x^{4} + z^{2}yw + zy^{2}w + z^{2}xw + y^{2}xw + zx^{2}w + yx^{2}w + zyw^{2} + zxw^{2} + yxw^{2} + x^{2}w^{2} q ↦ z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2} + y^{2}x^{2} + zx^{3} + yx^{3} + x^{4} + z^{2}yw + zy^{2}w + zx^{2}w + yx^{2}w + zyw^{2} + x^{2}w^{2} p ↦ z^{2}yx + zy^{2}x n ↦ z^{3}x+z^{2}yx+zy^{2}x+y^{3}x+z^{2}x^{2}+zyx^{2}+y^{2}x^{2}+zx^{3}+yx^{3}+z^{2}xw+y^{2}xw+zxw^{2}+yxw^{2} m ↦ z^{2}x^{2} + y^{2}x^{2} + x^{4} + z^{2}w^{2} + y^{2}w^{2} + w^{4} k ↦ z^{4}x + z^{3}yx + zy^{3}x + y^{4}x + z^{2}yx^{2} + zy^{2}x^{2} + zx^{4} + yx^{4} + z^{3}xw + z^{2}yxw + zy^{2}xw + y^{3}xw + z^{2}x^{2}w + y^{2}x^{2}w + z^{2}xw^{2} + y^{2}xw^{2} + zx^{2}w^{2} + yx^{2}w^{2} j ↦ z^{5}x + z^{4}yx + zy^{4}x + y^{5}x + z^{2}yx^{3} + zy^{2}x^{3} + z^{2}x^{4} + y^{2}x^{4} + zx^{5} + yx^{5} + z^{4}yw + z^{3}y^{2}w + z^{2}y^{3}w + zy^{4}w + z^{4}xw + y^{4}xw + z^{2}x^{3}w + y^{2}x^{3}w + z^{3}yw^{2} + zy^{3}w^{2} + z^{3}xw^{2} + z^{2}yxw^{2} + zy^{2}xw^{2} + y^{3}xw^{2} + zx^{3}w^{2} + yx^{3}w^{2}

Restriction from P̂ to Ê_{2}: z ↦ 0 y ↦ 0 x ↦ y w ↦ 0 v ↦ z^{2} + zy + yx + x^{2} u ↦ 0 t ↦ 0 s ↦ 0 r ↦ z^{2}x^{2} + zyx^{2} + y^{2}x^{2} + yx^{3} + x^{4} + z^{2}w^{2} + zyw^{2} + y^{2}w^{2} + yxw^{2} + x^{2}w^{2} q ↦ z^{2}x^{2} + zyx^{2} + yx^{3} + x^{4} + z^{2}w^{2} + zyw^{2} + yxw^{2} + x^{2}w^{2} p ↦ z^{2}yx + zy^{2}x + y^{2}x^{2} + yx^{3} + z^{2}yw + zy^{2}w + y^{2}xw + yx^{2}w n ↦ z^{2}yx + zy^{2}x + z^{2}x^{2} + zyx^{2} m ↦ x^{4} + w^{4} k ↦ 0 j ↦ z^{2}y^{2}x^{2} +zy^{3}x^{2} +y^{4}x^{2} +z^{2}yx^{3} +zy^{2}x^{3} +y^{3}x^{3} +yx^{5} +z^{2}yx^{2}w +zy^{2}x^{2}w +y^{2}x^{3}w + yx^{4}w +z^{2}y^{2}w^{2} +zy^{3}w^{2} +y^{4}w^{2} +z^{2}yxw^{2} +zy^{2}xw^{2} +y^{3}xw^{2} +yx^{3}w^{2} +z^{2}yw^{3} +zy^{2}w^{3} + y^{2}xw^{3} + yx^{2}w^{3}

Restriction from P̂ to Ê_{3}: z ↦ 0 y ↦ 0 x ↦ z w ↦ zy v ↦ zy + y^{2} u ↦ 0 t ↦ zy^{2} s ↦ zy^{3} + zyx^{2} + zyw r ↦ zy^{3} + y^{4} + zyx^{2} + y^{2}x^{2} + zyw + y^{2}w q ↦ y^{4} + y^{2}x^{2} + y^{2}w p ↦ zy^{3} + zy^{2}x + zyw n ↦ zy^{3} + zyx^{2} + zyw + y^{2}w + w^{2} m ↦ y^{4} + x^{4} + w^{2} k ↦ zy^{4} + zy^{2}x^{2} + zy^{2}w j ↦ zy^{4}x + zy^{2}x^{3} + zy^{2}xw + zyx^{2}w

Restriction from P̂ to Ê_{4}: z ↦ 0 y ↦ z x ↦ y w ↦ 0 v ↦ zy u ↦ 0 t ↦ 0 s ↦ zy^{3} + zyx^{2} + zyw r ↦ zy^{3} + y^{4} + zyx^{2} + y^{2}x^{2} + zyw + y^{2}w q ↦ zy^{3} + zyx^{2} + zyw p ↦ zy^{3} + zy^{2}x + zyw n ↦ zy^{3} + zyx^{2} + zyw + y^{2}w + w^{2} m ↦ y^{4} + x^{4} + w^{2} k ↦ zy^{2}w + zw^{2} j ↦ y^{6} + zy^{4}x + y^{4}x^{2} + zy^{2}x^{3} + y^{4}w + zy^{2}xw + zyx^{2}w

Restriction from P̂ to Ê_{5}: z ↦ 0 y ↦ z + y x ↦ y w ↦ z^{2} v ↦ y^{2} u ↦ zy^{2} t ↦ zy^{2} s ↦ zy^{2}x r ↦ zy^{2}x q ↦ zyx^{2} p ↦ zy^{2}x n ↦ zy^{2}x + w m ↦ x^{4} + w k ↦ z^{2}yx^{2} + zw + yw j ↦ zy^{2}x^{3}

Restriction from 2M_{12} to Ê_{1}: y ↦ 0 z ↦ z^{2}y + zy^{2} + z^{2}x + y^{2}x + zx^{2} + yx^{2} + z^{2}w + y^{2}w + zw^{2} + yw^{2} β ↦ z^{4} + z^{2}y^{2} + y^{4} + z^{2}x^{2} + y^{2}x^{2} + x^{4} + z^{2}yw + zy^{2}w + z^{2}xw + y^{2}xw + zx^{2}w + yx^{2}w + z^{2}w^{2} + zyw^{2} + y^{2}w^{2} + zxw^{2} + yxw^{2} + x^{2}w^{2} + w^{4} τ ↦ x^{4} + z^{2}yw + zy^{2}w + z^{2}xw + y^{2}xw + zx^{2}w + yx^{2}w + zyw^{2} + zxw^{2} + yxw^{2} + x^{2}w^{2} γ ↦ z^{4}y + zy^{4} + z^{4}x + y^{4}x + zx^{4} + yx^{4} + z^{4}w + y^{4}w + zw^{4} + yw^{4} δ ↦ z^{4}yw +zy^{4}w +z^{4}xw +y^{4}xw +zx^{4}w +yx^{4}w +z^{2}y^{2}w^{2} +z^{2}x^{2}w^{2} +y^{2}x^{2}w^{2} +x^{4}w^{2} + zyw^{4} + zxw^{4} + yxw^{4} + x^{2}w^{4} ϵ ↦ z^{4}y^{2}w + z^{2}y^{4}w + z^{4}x^{2}w + y^{4}x^{2}w + z^{2}x^{4}w + y^{2}x^{4}w + z^{4}yw^{2} + zy^{4}w^{2} + z^{4}xw^{2} + y^{4}xw^{2} + zx^{4}w^{2} + yx^{4}w^{2} + z^{2}yw^{4} + zy^{2}w^{4} + z^{2}xw^{4} + y^{2}xw^{4} + zx^{2}w^{4} + yx^{2}w^{4} θ ↦ z^{2}yx + zy^{2}x + z^{2}x^{2} + y^{2}x^{2} + zx^{3} + yx^{3} + z^{2}xw + y^{2}xw + zxw^{2} + yxw^{2} ϕ ↦ z^{4}yx + zy^{4}x + z^{4}x^{2} + y^{4}x^{2} + zx^{5} + yx^{5} + z^{4}xw + y^{4}xw + zxw^{4} + yxw^{4} ξ ↦ z^{2}yx^{2} + zy^{2}x^{2} + z^{2}x^{3} + y^{2}x^{3} + zx^{4} + yx^{4} + z^{2}x^{2}w + y^{2}x^{2}w + zx^{2}w^{2} + yx^{2}w^{2} η ↦ z^{4}yx^{2} + zy^{4}x^{2} + z^{4}x^{3} + y^{4}x^{3} + zx^{6} + yx^{6} + z^{4}x^{2}w + y^{4}x^{2}w + zx^{2}w^{4} + yx^{2}w^{4} ψ ↦ z^{4}y^{2}x^{2}w + z^{2}y^{4}x^{2}w + z^{4}x^{4}w + y^{4}x^{4}w + z^{2}x^{6}w + y^{2}x^{6}w + z^{4}yx^{2}w^{2} + zy^{4}x^{2}w^{2} + z^{4}x^{3}w^{2} + y^{4}x^{3}w^{2} + zx^{6}w^{2} + yx^{6}w^{2} + z^{2}yx^{2}w^{4} + zy^{2}x^{2}w^{4} + z^{2}x^{3}w^{4} + y^{2}x^{3}w^{4} + zx^{4}w^{4} + yx^{4}w^{4} χ ↦ z^{6}yw+z^{5}y^{2}w+z^{4}y^{3}w+z^{3}y^{4}w+z^{2}y^{5}w+zy^{6}w+z^{6}xw+y^{6}xw+z^{5}x^{2}w+z^{4}yx^{2}w+ zy^{4}x^{2}w+y^{5}x^{2}w+z^{3}x^{4}w+y^{3}x^{4}w+z^{2}x^{5}w+y^{2}x^{5}w+z^{5}yw^{2}+z^{3}y^{3}w^{2}+zy^{5}w^{2}+z^{5}xw^{2}+ z^{4}yxw^{2}+z^{3}y^{2}xw^{2}+z^{2}y^{3}xw^{2}+zy^{4}xw^{2}+y^{5}xw^{2}+z^{4}x^{2}w^{2}+z^{3}yx^{2}w^{2}+zy^{3}x^{2}w^{2}+y^{4}x^{2}w^{2}+ z^{3}x^{3}w^{2} + z^{2}yx^{3}w^{2} + zy^{2}x^{3}w^{2} + y^{3}x^{3}w^{2} + zyx^{4}w^{2} + zx^{5}w^{2} + yx^{5}w^{2} + z^{4}yw^{3} + z^{3}y^{2}w^{3} + z^{2}y^{3}w^{3} + zy^{4}w^{3} + z^{4}xw^{3} + y^{4}xw^{3} + z^{3}x^{2}w^{3} + z^{2}yx^{2}w^{3} + zy^{2}x^{2}w^{3} + y^{3}x^{2}w^{3} + z^{3}yw^{4} + zy^{3}w^{4}+z^{3}xw^{4}+z^{2}yxw^{4}+zy^{2}xw^{4}+y^{3}xw^{4}+z^{2}x^{2}w^{4}+zyx^{2}w^{4}+y^{2}x^{2}w^{4}+zx^{3}w^{4}+yx^{3}w^{4}+ x^{4}w^{4} +z^{2}yw^{5} +zy^{2}w^{5} +z^{2}xw^{5} +y^{2}xw^{5} +zx^{2}w^{5} +yx^{2}w^{5} +zyw^{6} +zxw^{6} +yxw^{6} +x^{2}w^{6} ρ ↦ z^{2}yx^{3} + zy^{2}x^{3} + z^{2}x^{4} + y^{2}x^{4} + zx^{5} + yx^{5} + z^{2}x^{3}w + y^{2}x^{3}w + zx^{3}w^{2} + yx^{3}w^{2} ω ↦ z^{4}yx^{3} + zy^{4}x^{3} + z^{4}x^{4} + y^{4}x^{4} + zx^{7} + yx^{7} + z^{4}y^{2}xw + z^{2}y^{4}xw + z^{2}x^{5}w + y^{2}x^{5}w + z^{4}yxw^{2} + zy^{4}xw^{2} + z^{4}x^{2}w^{2} + y^{4}x^{2}w^{2} + zx^{5}w^{2} + yx^{5}w^{2} + z^{2}yxw^{4} + zy^{2}xw^{4} + z^{2}x^{2}w^{4} + y^{2}x^{2}w^{4}

Restriction from 2M_{12} to Ê_{3}: y ↦ 0 z ↦ 0 β ↦ zy^{2}x + zyx^{2} + y^{4} + y^{2}x^{2} + x^{4} τ ↦ zy^{2}x + zyx^{2} + y^{4} + y^{2}x^{2} + w^{2} γ ↦ 0 δ ↦ zy^{4}x + zyx^{4} + y^{4}x^{2} + y^{2}x^{4} ϵ ↦ zy^{4}x^{2} + zy^{2}x^{4} θ ↦ zy^{2}x + zyx^{2} ϕ ↦ zy^{4}x + zyx^{4} ξ ↦ 0 η ↦ zy^{4}x^{2} + zy^{2}x^{4} ψ ↦ zy^{4}x^{4} + zy^{4}x^{2}w + zy^{2}x^{6} + zy^{2}x^{4}w χ ↦ zy^{4}x^{3} + zy^{4}xw + zyx^{6} + zyx^{4}w + y^{4}x^{4} + y^{4}x^{2}w + y^{2}x^{6} + y^{2}x^{4}w ρ ↦ zy^{4}x + zy^{2}x^{3} + zy^{2}xw + zyx^{2}w ω ↦ zy^{6}x + zy^{5}x^{2} + zy^{4}xw + zy^{3}x^{4} + zy^{2}x^{5} + zyx^{4}w

Restriction from 2M_{12} to Ê_{5}: y ↦ zy^{2} z ↦ 0 β ↦ x^{4} τ ↦ w γ ↦ 0 δ ↦ 0 ϵ ↦ 0 θ ↦ 0 ϕ ↦ 0 ξ ↦ 0 η ↦ 0 ψ ↦ zy^{2}x^{6} χ ↦ zy^{2}x^{5} ρ ↦ zy^{2}x^{3} ω ↦ zy^{2}x^{5}

References

1. A. Adem, J. Maginnis, and R. J. Milgram, The geometry and cohomology of the Mathieu group M_{12}, J. Algebra 139 (1991), 90–133.

2. D. J. Benson and J. F. Carlson, Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. 132 (1994), 447–488.

3. W. Bosma and J. Cannon, Handbook of Magma Functions, Magma Computer Algebra, Sydney, 1996.

4. J. F. Carlson, Calculating group cohomology: Tests for completion, J. Symb. Comp., to appear.

5. , Problems in the computation of group cohomology, Proceedings of the Euroconference on Computational Methods for Representations of Groups and Algebras, to appear.

6. J. F. Carlson, E. Green, and G. J. A. Schneider, Computing the ext algebras for the group algebras of finite groups, J. Symb. Comp. 24 (1997), 317–325.

7. J. F. Carlson, J. Maginnis, and R. J. Milgram, The cohomology of the sporadic groups J_{2} and J_{3}, J. Algebra 214 (1999), 143–173.

8. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Simple Groups, Oxford University Press, 1985.

9. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften, vol. 290, Springer-Verlag, Berlin/New York, 1988.

10. J. Duflot, Depth and equivariant cohomology, Comment. Math. Helvetici 56 (1981), 617–637.

11. D. G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549–572.

12. , The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573–602.

13. J.-P. Serre, Homologie singulière des espaces fibrés, Ann. of Math. 54 (3) (1951), 425–505.

Dept. of Mathematics, University of Georgia, Athens GA 30602, USA

   E-mail address: djb@byrd.math.uga.edu

Dept. of Mathematics, University of Georgia, Athens GA 30602, USA

   E-mail address: jfc@sloth.math.uga.edu