Cohomology of group number 8 of order 81

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General information on the group

  • The group has 2 minimal generators and exponent 9.
  • 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t4  −  t3  +  t2  +  1
    (t  −  1)2 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-3,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 7:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. a_2_2, a nilpotent element of degree 2
  5. b_2_0, an element of degree 2
  6. a_3_2, a nilpotent element of degree 3
  7. a_4_1, a nilpotent element of degree 4
  8. b_4_2, an element of degree 4
  9. a_5_2, a nilpotent element of degree 5
  10. a_5_3, a nilpotent element of degree 5
  11. b_6_3, an element of degree 6
  12. c_6_4, a Duflot regular element of degree 6
  13. a_7_5, a nilpotent element of degree 7

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Ring relations

There are 6 "obvious" relations:
   a_1_02, a_1_12, a_3_22, a_5_22, a_5_32, a_7_52

Apart from that, there are 59 minimal relations of maximal degree 13:

  1. a_1_0·a_1_1
  2. a_2_1·a_1_0
  3. a_2_2·a_1_1
  4. a_2_2·a_1_0 − a_2_1·a_1_1
  5. b_2_0·a_1_1
  6. a_2_12
  7. a_2_22
  8. a_2_1·a_2_2
  9. a_2_2·b_2_0
  10. a_2_1·b_2_0
  11. a_1_1·a_3_2
  12. a_1_0·a_3_2
  13. a_2_2·a_3_2
  14. b_2_0·a_3_2
  15. a_2_1·a_3_2
  16. a_4_1·a_1_1
  17. a_4_1·a_1_0
  18. b_4_2·a_1_0
  19. a_2_2·a_4_1
  20. a_2_1·a_4_1
  21. b_2_0·b_4_2 − b_2_0·a_4_1
  22. a_2_1·b_4_2
  23. a_1_1·a_5_2
  24.  − b_2_0·a_4_1 + a_1_0·a_5_2
  25.  − a_2_2·b_4_2 + a_1_1·a_5_3
  26. b_2_0·a_4_1 + a_1_0·a_5_3
  27. a_4_1·a_3_2
  28. a_2_2·a_5_2
  29. a_2_1·a_5_2
  30. a_2_2·a_5_3
  31. b_2_0·a_5_3 + b_2_0·a_5_2
  32. a_2_1·a_5_3
  33. b_6_3·a_1_1 + b_4_2·a_3_2
  34. b_6_3·a_1_0
  35. a_4_12
  36. a_3_2·a_5_2
  37.  − a_4_1·b_4_2 + a_3_2·a_5_3
  38. a_4_1·b_4_2 + a_2_2·b_6_3
  39. b_2_0·b_6_3 − b_2_0·a_1_0·a_5_2
  40. a_2_1·b_6_3
  41. a_4_1·b_4_2 + a_1_1·a_7_5
  42. a_1_0·a_7_5 + b_2_0·a_1_0·a_5_2
  43. b_4_2·a_5_2 + b_4_22·a_1_1
  44. a_4_1·a_5_2
  45. a_4_1·a_5_3
  46. b_6_3·a_3_2 − b_4_22·a_1_1
  47. a_2_2·a_7_5 + a_2_1·c_6_4·a_1_1
  48. b_2_0·a_7_5 + b_2_02·a_5_2 + b_2_0·c_6_4·a_1_0
  49. a_2_1·a_7_5 + a_2_1·c_6_4·a_1_1
  50. a_5_2·a_5_3 + b_4_2·a_1_1·a_5_3
  51. a_4_1·b_6_3 + a_5_2·a_5_3
  52. a_5_2·a_5_3 + a_3_2·a_7_5
  53. b_6_3·a_5_2 − b_4_22·a_3_2
  54.  − b_6_3·a_5_3 + b_4_2·a_7_5 − b_4_22·a_3_2 + b_4_2·c_6_4·a_1_1
  55. a_4_1·a_7_5
  56. b_6_32 + b_4_23 + a_5_3·a_7_5 − c_6_4·a_1_1·a_5_3 + c_6_4·a_1_0·a_5_2
  57.  − b_6_32 − b_4_23 + a_5_2·a_7_5 − c_6_4·a_1_0·a_5_2
  58. b_6_32 + b_4_23 + b_4_2·a_1_1·a_7_5
  59. b_6_3·a_7_5 + b_4_22·a_5_3 − b_4_23·a_1_1 − b_4_2·c_6_4·a_3_2

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 13.
  • 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_6_4, a Duflot regular element of degree 6
    2.  − b_4_23 + b_2_06, an element of degree 12
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 16].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 1 elements of degree 4.

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Restriction maps

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. b_2_0 → 0, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_4_1 → 0, an element of degree 4
  8. b_4_2 → 0, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. a_5_3 → 0, an element of degree 5
  11. b_6_3 → 0, an element of degree 6
  12. c_6_4 → c_2_03, an element of degree 6
  13. a_7_5 → 0, an element of degree 7

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

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. b_2_0 → c_2_2, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
  8. b_4_2 → c_2_2·a_1_0·a_1_1, an element of degree 4
  9. a_5_2 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  10. a_5_3 → c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  11. b_6_3 → c_2_22·a_1_0·a_1_1, an element of degree 6
  12. c_6_4 →  − c_2_22·a_1_0·a_1_1 − c_2_1·c_2_22 + c_2_13, an element of degree 6
  13. a_7_5 → c_2_23·a_1_0 − c_2_13·a_1_1, an element of degree 7

Restriction map to a maximal el. ab. subgp. 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_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. b_2_0 → 0, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_4_1 → 0, an element of degree 4
  8. b_4_2 →  − c_2_22, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. a_5_3 → c_2_22·a_1_1, an element of degree 5
  11. b_6_3 → c_2_23, an element of degree 6
  12. c_6_4 → c_2_23 − c_2_1·c_2_22 + c_2_13, an element of degree 6
  13. a_7_5 →  − c_2_23·a_1_1, an element of degree 7

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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