Mod-2-Cohomology of SymplecticGroup(4,3), a group of order 51840

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General information on the group

  • SymplecticGroup(4,3) is a group of order 51840.
  • The group order factors as 27 · 34 · 5.
  • The group is defined by Group([(1,2)(3,5)(4,7)(6,10)(8,12)(9,13)(11,16)(14,20)(15,21)(17,24)(18,25)(19,27)(23,31)(28,32)(29,37)(30,39)(33,38)(34,43)(35,45)(41,49)(46,54)(47,51)(48,57)(50,53)(52,61)(55,56)(58,67)(59,60)(62,70)(65,71)(66,72)(68,74)(69,75)(73,76)(77,79)(78,80),(1,3,6)(2,4,8)(5,9,14,12,18,26,34,44,25)(7,11,17,10,15,22,30,40,21)(13,19,28,36,47,56,20,29,38)(16,23,32,42,51,60,24,33,37)(27,35,46,55,65,67,73,57,66)(31,41,50,59,68,70,76,61,69)(39,48,58)(43,52,62)(45,53,63)(49,54,64)(71,77,75)(72,74,78)]).
  • It is non-abelian.
  • It has 2-Rank 2.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.


Structure of the cohomology ring

The computation was based on 17 stability conditions for H*(Syl2(Sp4(3)); GF(2)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    (1  −  t  +  t2) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6)

    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Ring generators

The cohomology ring has 4 minimal generators of maximal degree 8:

  1. a_3_0, a nilpotent element of degree 3
  2. b_4_0, an element of degree 4
  3. a_7_0, a nilpotent element of degree 7
  4. c_8_1, a Duflot element of degree 8

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Ring relations

There are 2 minimal relations of maximal degree 14:

  1. a_3_02
  2. a_7_02


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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 14 using the Hilbert-Poincaré criterion.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_1, an element of degree 8
    2. b_4_0, an element of degree 4
  • A Duflot regular sequence is given by c_8_1.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].


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Restriction maps

Expressing the generators as elements of H*(Syl2(Sp4(3)); GF(2))

  1. a_3_0a_2_4·a_1_2 + a_1_23 + a_1_13
  2. b_4_0b_1_04 + b_4_8
  3. a_7_0a_2_4·a_5_10 + a_2_4·a_2_5·b_1_03 + a_1_22·a_5_10 + a_1_12·a_5_10 + a_1_12·a_5_9
       + b_4_8·a_1_23 + b_4_8·a_1_1·a_1_22
  4. c_8_1b_4_82 + a_2_4·a_2_5·b_1_04 + a_2_42·b_1_04 + a_2_42·a_2_5·b_1_02
       + a_1_23·a_5_10 + a_1_13·a_5_10 + a_1_13·a_5_9 + b_4_8·a_1_12·a_1_22
       + b_4_8·a_1_13·a_1_2 + c_8_15

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_3_00, an element of degree 3
  2. b_4_00, an element of degree 4
  3. a_7_00, an element of degree 7
  4. c_8_1c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_00, an element of degree 3
  2. b_4_0c_1_14, an element of degree 4
  3. a_7_00, an element of degree 7
  4. c_8_1c_1_04·c_1_14 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_00, an element of degree 3
  2. b_4_0c_1_14, an element of degree 4
  3. a_7_00, an element of degree 7
  4. c_8_1c_1_04·c_1_14 + c_1_08, an element of degree 8


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Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2



Last change: 14.12.2010