Cohomology of group number 937 of order 128

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General information on the group

  • The group is also known as Syl2(Sp4(3)), the Sylow 2-subgroup of Symplectic Group Sp_4(3).
  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

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
    (t2  +  t  +  1) · (t6  +  2·t4  −  t3  +  2·t2  +  1)
    (t  −  1)2 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 8:

  1. a_1_1, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_0, an element of degree 1
  4. a_2_4, a nilpotent element of degree 2
  5. a_2_5, a nilpotent element of degree 2
  6. b_4_8, an element of degree 4
  7. a_5_9, a nilpotent element of degree 5
  8. a_5_10, a nilpotent element of degree 5
  9. c_8_15, a Duflot regular element of degree 8

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Ring relations

There are 21 minimal relations of maximal degree 10:

  1. a_1_1·b_1_0
  2. a_1_2·b_1_0
  3. a_2_4·a_1_1 + a_1_1·a_1_22 + a_1_12·a_1_2 + a_1_13
  4. a_2_5·a_1_1 + a_2_4·a_1_2 + a_1_1·a_1_22 + a_1_12·a_1_2 + a_1_13
  5. a_2_5·a_1_2
  6. a_1_14
  7. a_1_24
  8. a_2_52 + a_2_4·a_2_5 + a_2_42 + a_1_12·a_1_22 + a_1_13·a_1_2
  9. b_4_8·b_1_0
  10. a_2_43
  11. b_1_0·a_5_9 + a_2_5·b_1_04 + a_2_4·b_1_04 + a_2_42·b_1_02
  12. a_1_2·a_5_9 + a_1_1·a_5_10 + a_1_1·a_5_9 + b_4_8·a_1_22
  13. b_1_0·a_5_10 + a_2_5·b_1_04 + a_2_4·a_2_5·b_1_02 + a_2_42·b_1_02
  14. a_2_5·a_5_10 + a_2_5·a_5_9 + a_2_4·b_4_8·a_1_2 + a_2_4·a_2_5·b_1_03 + b_4_8·a_1_23
       + b_4_8·a_1_1·a_1_22 + b_4_8·a_1_12·a_1_2 + b_4_8·a_1_13 + a_2_42·a_2_5·b_1_0
  15. a_2_5·a_5_9 + a_2_4·a_5_10 + a_2_4·b_4_8·a_1_2 + a_2_4·a_2_5·b_1_03 + a_2_42·b_1_03
  16. a_2_4·a_5_9 + a_2_4·a_2_5·b_1_03 + a_2_42·b_1_03 + a_1_1·a_1_2·a_5_10
       + a_1_12·a_5_9 + b_4_8·a_1_23 + b_4_8·a_1_13
  17. a_2_42·b_4_8 + a_1_13·a_5_9 + b_4_8·a_1_12·a_1_22
  18. a_2_4·a_2_5·b_4_8 + a_1_23·a_5_10 + a_1_1·a_1_22·a_5_10 + a_1_12·a_1_2·a_5_10
       + a_1_13·a_5_10 + b_4_8·a_1_12·a_1_22 + b_4_8·a_1_13·a_1_2
  19. a_2_4·b_4_82 + a_5_92 + b_4_8·a_1_1·a_5_9 + b_4_82·a_1_22 + b_4_82·a_1_1·a_1_2
       + b_4_82·a_1_12 + a_2_4·a_2_5·b_1_06 + a_1_13·a_1_22·a_5_10 + c_8_15·a_1_12
  20. a_2_5·b_4_82 + a_2_4·b_4_82 + a_5_9·a_5_10 + a_5_92 + b_4_8·a_1_2·a_5_10
       + b_4_82·a_1_22 + b_4_82·a_1_1·a_1_2 + a_2_4·a_2_5·b_1_06 + a_2_42·b_1_06
       + a_1_13·a_1_22·a_5_10 + c_8_15·a_1_1·a_1_2
  21. a_2_5·b_4_82 + a_5_102 + a_5_92 + b_4_8·a_1_2·a_5_10 + b_4_8·a_1_1·a_5_10
       + b_4_8·a_1_1·a_5_9 + b_4_82·a_1_22 + b_4_82·a_1_1·a_1_2 + b_4_82·a_1_12
       + a_2_42·b_1_06 + a_1_13·a_1_22·a_5_10 + c_8_15·a_1_22

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 11.
  • However, the last relation was already found in degree 10 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_15, a Duflot regular element of degree 8
    2. b_1_04 + b_4_8, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. a_5_9 → 0, an element of degree 5
  8. a_5_10 → 0, an element of degree 5
  9. c_8_15 → c_1_08, an element of degree 8

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

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → c_1_1, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. a_5_9 → 0, an element of degree 5
  8. a_5_10 → 0, an element of degree 5
  9. c_8_15 → c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_4_8 → c_1_14, an element of degree 4
  7. a_5_9 → 0, an element of degree 5
  8. a_5_10 → 0, an element of degree 5
  9. c_8_15 → c_1_18 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009