Cohomology of group number 20 of order 243

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General information on the group

  • The group has 2 minimal generators and exponent 27.
  • 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_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_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_0·a_1_0
  3. a_2_1·a_1_1 + a_2_0·a_1_1
  4. a_2_1·a_1_0 − a_2_0·a_1_1
  5. b_2_2·a_1_0
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9. a_2_0·b_2_2
  10. a_2_1·b_2_2
  11. a_1_1·a_3_2
  12. a_1_0·a_3_2
  13. b_2_2·a_3_2
  14. a_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_1 + b_4_2·a_1_0
  19. a_2_0·a_4_1
  20. a_2_1·a_4_1
  21. b_2_2·b_4_2
  22. a_2_1·b_4_2
  23.  − b_2_2·a_4_1 + a_1_1·a_5_2
  24. a_1_0·a_5_2
  25. b_2_2·a_4_1 + a_2_0·b_4_2 + a_1_1·a_5_3
  26.  − a_2_0·b_4_2 + a_1_0·a_5_3
  27. a_4_1·a_3_2
  28. a_2_0·a_5_2
  29. a_2_1·a_5_2
  30. b_2_2·a_5_3 + b_2_2·a_5_2
  31. a_2_0·a_5_3
  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 + b_4_2·a_3_2
  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. b_2_2·b_6_3 − b_2_2·a_1_1·a_5_2
  39. a_4_1·b_4_2 + a_2_0·b_6_3
  40. a_2_1·b_6_3
  41.  − a_4_1·b_4_2 + a_1_1·a_7_5 + b_2_2·a_1_1·a_5_2
  42. a_4_1·b_4_2 + a_1_0·a_7_5
  43. b_4_2·a_5_2 + b_4_22·a_1_0
  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_0
  47. b_2_2·a_7_5 + b_2_22·a_5_2
  48. a_2_0·a_7_5
  49. a_2_1·a_7_5
  50. a_5_2·a_5_3 + b_4_2·a_1_0·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
  55. a_4_1·a_7_5
  56. a_5_3·a_7_5
  57. b_6_32 − b_4_23 + a_5_2·a_7_5
  58.  − b_6_32 + b_4_23 + b_4_2·a_1_0·a_7_5
  59. b_6_3·a_7_5 − b_4_22·a_5_3

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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_2 + b_2_22, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 8].
  • 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_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_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 → 0, an element of degree 1
  2. a_1_1 → a_1_1, 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_2_2, 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_1·c_2_22 − c_2_13, 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 → 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_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

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243


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