Cohomology of group number 31 of order 128

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General information on the group

  • The group has 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t5  −  t3  −  t2  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. a_3_1, a nilpotent element of degree 3
  7. a_3_3, a nilpotent element of degree 3
  8. a_3_4, a nilpotent element of degree 3
  9. b_3_2, an element of degree 3
  10. a_4_3, a nilpotent element of degree 4
  11. a_4_4, a nilpotent element of degree 4
  12. a_4_5, a nilpotent element of degree 4
  13. c_4_6, a Duflot regular element of degree 4
  14. c_4_7, a Duflot regular element of degree 4

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Ring relations

There are 65 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_0
  5. a_2_1·a_1_1
  6. a_2_1·a_1_0 + a_2_0·a_1_1
  7. b_2_2·a_1_1
  8. b_2_2·a_1_0 + a_2_0·a_1_1
  9. a_2_02
  10. a_2_0·a_2_1
  11. a_2_12
  12. a_2_1·b_2_2 + a_2_0·b_2_2 + a_1_1·a_3_1
  13. a_1_0·a_3_1
  14. a_2_0·b_2_2 + a_1_1·a_3_3
  15. a_2_0·b_2_2 + a_1_0·a_3_3
  16. a_2_1·b_2_2 + a_1_1·a_3_4
  17. a_2_1·b_2_2 + a_2_0·b_2_2 + a_1_0·a_3_4
  18. a_1_1·b_3_2 + a_2_1·b_2_2
  19. a_1_0·b_3_2
  20. a_2_1·a_3_1 + a_2_0·a_3_1
  21. a_2_0·a_3_3
  22. b_2_2·a_3_3 + b_2_2·a_3_1 + a_2_1·a_3_3 + a_2_0·a_3_1
  23. b_2_2·a_3_3 + b_2_2·a_3_1 + a_2_0·a_3_4 + a_2_0·a_3_1
  24. b_2_2·a_3_3 + b_2_2·a_3_1 + a_2_1·a_3_4
  25. a_2_0·b_3_2 + a_2_0·a_3_1
  26. a_2_1·b_3_2
  27. a_4_3·a_1_1 + a_2_0·a_3_1
  28. a_4_3·a_1_0
  29. b_2_2·a_3_3 + b_2_2·a_3_1 + a_4_4·a_1_1
  30. a_4_4·a_1_0 + a_2_0·a_3_1
  31. b_2_2·a_3_3 + b_2_2·a_3_1 + a_4_5·a_1_1
  32. b_2_2·a_3_3 + b_2_2·a_3_1 + a_4_5·a_1_0 + a_2_0·a_3_1
  33. a_3_12
  34. a_3_32
  35. a_3_1·a_3_3
  36. a_3_3·a_3_4 + a_3_1·a_3_4
  37. a_3_42
  38. a_3_3·b_3_2 + a_3_1·b_3_2
  39. b_3_22 + b_2_23
  40. a_3_3·b_3_2 + b_2_2·a_4_3
  41. a_2_0·a_4_3
  42. a_2_1·a_4_3
  43. a_3_4·b_3_2 + b_2_2·a_4_4 + a_3_3·a_3_4
  44. a_2_0·a_4_4
  45. a_2_1·a_4_4
  46. b_2_2·a_4_5 + a_3_3·a_3_4
  47. a_2_0·a_4_5
  48. a_2_1·a_4_5
  49. a_4_3·a_3_3
  50. a_4_3·a_3_1 + a_2_0·c_4_6·a_1_1
  51. a_4_3·b_3_2 + b_2_22·a_3_1 + a_2_0·c_4_6·a_1_1
  52. a_4_4·a_3_3 + a_4_3·a_3_4
  53. a_4_4·a_3_1 + a_4_3·a_3_4
  54. a_4_4·b_3_2 + b_2_22·a_3_4 + a_4_3·a_3_4 + a_2_0·c_4_6·a_1_1
  55. a_4_4·a_3_4 + a_2_0·c_4_7·a_1_1
  56. a_4_5·a_3_3 + a_2_0·c_4_7·a_1_1 + a_2_0·c_4_6·a_1_1
  57. a_4_5·a_3_1 + a_2_0·c_4_6·a_1_1
  58. a_4_5·b_3_2 + a_4_3·a_3_4 + a_2_0·c_4_6·a_1_1
  59. a_4_5·a_3_4
  60. a_4_32
  61. a_4_3·a_4_4 + b_2_2·a_3_1·a_3_4
  62. a_4_42
  63. a_4_3·a_4_5
  64. a_4_4·a_4_5
  65. a_4_52

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_4_6, a Duflot regular element of degree 4
    2. c_4_7, a Duflot regular element of degree 4
    3. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. a_3_1 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_2 → 0, an element of degree 3
  10. a_4_3 → 0, an element of degree 4
  11. a_4_4 → 0, an element of degree 4
  12. a_4_5 → 0, an element of degree 4
  13. c_4_6 → c_1_04, an element of degree 4
  14. c_4_7 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_1_22, an element of degree 2
  6. a_3_1 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_2 → c_1_23, an element of degree 3
  10. a_4_3 → 0, an element of degree 4
  11. a_4_4 → 0, an element of degree 4
  12. a_4_5 → 0, an element of degree 4
  13. c_4_6 → c_1_24 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  14. c_4_7 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4

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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