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The Cohomologies of the Sylow 2-Subgroups of a Symplectic Group of Degree Six and of the Third Conway Group

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The Cohomologies of the Sylow 2-Subgroups of a Symplectic Group of Degree Six and of the Third Conway Group THE COHOMOLOGIES OF THE SYLOW 2-SUBGROUPS OF A SYMPLECTIC GROUP OF DEGREE SIX AND OF THE THIRD CONWAY GROUP JOHN MAGINNIS Abstract. The third Conway group Co3 is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2-subgroup S is computed, an important step in calculating the mod 2 cohomology of Co3. The spectral sequence for the central extension of S is described and collapses at the sixth page. Generators are described in terms of the Evens norm or transfers from subgroups. The central quotient S0 = S=2 is the Sylow 2-subgroup of the symplectic group Sp6(F2) of six by six matrices over the field of two elements. The cohomology of S0 is computed, and is detected by restriction to elementary abelian 2-subgroups. 1. Introduction A presentation for the Sylow 2-subgroup S of the sporadic finite simple group Co3 is given by Benson [1]. We will use a different presentation, with generators which can be described in terms of Benson's generators fa1; a2; a3; a4; b1; b2; c1; c2; eg as follows. w = c3; x = b2c1c2; y = ec3; z = a4; a = a3a4c2; b = a3a4b1c2; c = a3; t = a4c1; u = a2; v = a1: We have the following set of relations for the generators fw; x; y; z; a; b; c; t; u; vg. w2 = x2 = y2 = z2 = c2 = u2 = v2 = [w; y] = [w; z] = [w; c] = [w; v] = [x; z] = [x; u] = [x; v] = [y; c] = [y; u] = [y; v] = [a; z] = [c; z] = [u; z] = [v; z] = [c; t] = [c; u] = [c; v] = [a; u] = [a; v] = [b; c] = [b; t] = [b; u] = [b; v] = [t; u] = [t; v] = [u; v] = 1; tu = [a; y]; t = [w; b]; av = [w; x]; b = [x; y]; c = [y; z] u = b2 = [x; c] = [b; z] = [b; y] = [b; x]; uv = [t; x] = [a; b]; v = t2 = a2 = [w; u] = [a; c] = [t; z] = [a; w] = [t; w] = [t; a] = [t; y] = [a; x]: The group S has order 1024 and a center Z(S) of order two, generated by v = a1. The subgroup generated by v = a1; u = a2; c = a3 and z = a4 is elementary abelian, and the 4 quotient group S=2 is isomorphic to the Sylow 2-subgroup U4 of the general linear group Date: 21 October 2009. 1991 Mathematics Subject Classification. 20. 1 2 JOHN MAGINNIS 4 GL4(F2). One of the maximal subgroups of Co3 is 2 :A8 ; see Finkelstein [5]. Note that we are using the Atlas [3] notation for groups, where pn denotes an elementary abelian p-group of rank n, H:G denotes a group extension with normal subgroup H and quotient group G (with H : G the semidirect product), and p1+n = p:pn denotes an extraspecial p-group. The quotient by the center will be denoted S0 = S=hvi, which is the Sylow 2-subgroup of the symplectic group Sp6(F2); another maximal subgroup of Co3 is 2:Sp6(F2) [5]. We will also denote S00 = S0=hui = S=hv; ui;S000 = S=hv; u; ci and S0000 = S=hv; u; c; zi. We will denote various subgroups of S; S0;S00 et cetera by subscripts which denote generators not in the subgroup; for example Sx is the subgroup generated by hv; t; u; a; b; c; w; y; zi. This index two subgroup is the kernel of a group homomorphism S ! Z=2Z which represents a cohomology class of degree one that we will also denote as x. 0000 000 We have S ' U4 and S ' U4 ×2. We will determine the Lyndon-Hochschild-Serre spectral 00 0 00 0 sequences for the central extensions 2 ! S ! U4 × 2, 2 ! S ! S and 2 ! S ! S . We obtain the following. Theorem 1.1. The cohomology ring of S0 is generated by sixteen classes, with 4 degree one classes, 5 degree two classes, 4 degree three classes and 3 degree four classes. This cohomology ring contains no nilpotent elements. Theorem 1.2. The cohomology ring of S is generated by seventeen classes, with 4 degree one classes, 4 degree two classes, 2 degree three classes, 1 degree four class, 1 degree five class, 2 degree six classes, 2 degree seven classes, and 1 degree eight class. 0 Although this result for the Sylow 2-subgroup S of the symplectic group Sp6(F2) appears to be completely new, the latter result for the cohomology ring of S has also been obtained by Green and King [6] using a computer program and methods similar to those used by Carlson, et al [2]. The program computes a projective resolution (through some finite degree) and cohomology classes are represented as chain maps. Green and King determine the Poincare series for the cohomology ring of S (also verified in the present paper), show the depth of the ring is three, and compute the restrictions of the generators to representatives of the twenty conjugacy classes of maximal elementary abelian 2-subgroups. Green and King also list 78 relations for these 17 generators. The methods used in the present paper are quite different, and the computations were done mostly by hand. In addition, generators are described using the Evens norm [4] or transfers from subgroups. The webpage of Green and King does not give any details on the definition of their generators, so that we cannot describe our generators in terms of theirs. 2. The cohomology of S00 The cohomology of the group of unipotent upper triangular matrices U4 2 Syl2(GL4(F2)) was originally computed by Tezuka and Yagita [10], although a more complete description appears in the author's thesis [7] and in [8]. We will for the most part use the notation 2-COHOMOLOGY OF CO3 3 from these latter papers, although one generator, a degree three cohomology class D given as the Steenrod square of a degree two class, D = Sq1(d), will in this paper be replaced by a different generator Δ which can be described as a transfer from an index two subgroup. We will also write in place of d. Theorem 2.1. The cohomology ring of U4 is generated by eight classes, with 3 degree one classes w, x and y, 3 degree two classes , and , 1 degree three class Δ, and 1 degree four class T . We have the following additive description. ∗ H (U4; F2) = F2[x; ; ; T ](1; )(1; Δ)+wF2[w; ; T ]+yF2[y; ; T ]+wyF2[w; y; T ]+y F2[y; ; T ]: The relations are given by wx = xy = 0; w = y = w = y ; wΔ = yΔ = 0; 2 = xΔ + x2 + x2 + and Δ2 = x2T + x( + + )Δ + x2 + 2 + 2 + x2 . 4 The cohomology of U4 is detected on five elementary abelian 2-subgroups, 2 = ht; a; b; xi; 23 = ht; a; wi; 23 = ht; b; yi; 23 = ht; w; yi; and 23 = ht; ab; wyi. Proof. The above information appears in [10], [7] and [8]. The class Δ replaces the class D from [7] and [8], and we have Δ = D + x + y . The transfer map from the index 0000 two subgroup Sy = ht; a; b; w; xi satisfies T r(b) = x; T r() = + and T r(b) = Δ. 0000 The subgroup Sy ⊆ U4 is isomorphic to the central quotient U4=2 and has cohomology ring generated by 3 degree one classes w; x and b and 3 degree two classes ; and , with relations 2 2 2 ∗ wx = wb = w = 0 and = bx + x + b . The classes and T in H (U4; F2) are the ∗ Evens norm of b and , and the classes and 2 H (U4=2; F2) are in turn Evens norms from an elementary abelian subgroup of order 16. 2 4 2 2 Let us also remark that U4 is isomorphic to the wreath product 2 o 2 = 2 : 2 , with 2 ⊆ S4, 2 4 and the central quotient U4=2 is isomorphic to the wreath product 2 o 2 = 2 : 2. The cohomology of a wreath product can be computed using a theorem of Nakaoka proven in his paper on the cohomology of symmetric groups [9]. Also, a notation such as R(a; b; c) (used in the additive descriptions of cohomology rings) denotes a module, free over the ring R, generated by the elements a; b and c. □ Theorem 2.2. The cohomology ring of the group S00 is generated by thirteen classes, with 4 degree one classes w; x; y; z, 5 degree two classes ; ; ; ; K, 2 degree three classes Δ;L and 2 degree four classes T;M. The relations are given below. wx = xy = yz = 0; w = y = w = y ; wΔ = yΔ = 0 2 = xΔ + x2 + x2 + ; Δ2 = x2T + x( + + )Δ + x2 + 2 + 2 + x2 K2 = xzK + z2 + x2 ; L2 = z( + )L + z2T + (xΔ + x2 + x2 + 2 + ) M 2 = xz(+ )M+z (+ )L+x (+ )Δ+xzKT +(x2 +z2 )T + (xΔ+x2+x2 + 2+ ) 4 JOHN MAGINNIS xL = zΔ + ( + )(K + xz);KΔ = xM + z ( + ) KL = zM + x ( + ); ΔL = ( + )M + xzT; KM = xzM + z L + x Δ ΔM = x( + )M + ( + )L + xKT; LM = z( + )M + ( + )Δ + zKT yK = yL = yM = wK = wM = 0: 000 Proof. The group S is isomorphic to the product U4 ×2, and we shall denote by z the degree one cohomology class corresponding to the direct factor 2 (there is a group homomorphism 00 U4 × 2 ! Z=2Z with kernel the U4). The spectral sequence of the extension 2 ! S ! U4 × 2 has d2 differential determined by d2(c) = yz; obtained by restricting to the various subgroups: 26 = ht; a; b; c; x; zi; 25 = ht; a; c; w; zi; 2 2 2 D8 × 2 = ht; b; c; y; zi;D8 × 2 = ht; c; w; y; zi, and D8 × 2 = ht; ab; c; wy; zi.
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