Cohomology of group number 18 of order 243

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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 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
     − 1
    (t  −  1)3 · (t2  +  t  +  1)
  • 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 16 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. c_2_2, a Duflot regular element of degree 2
  6. a_3_2, 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. a_4_4, a nilpotent element of degree 4
  10. b_4_5, an element of degree 4
  11. a_5_7, a nilpotent element of degree 5
  12. a_5_8, a nilpotent element of degree 5
  13. a_6_7, a nilpotent element of degree 6
  14. b_6_9, an element of degree 6
  15. c_6_11, a Duflot regular element of degree 6
  16. a_7_14, a nilpotent element of degree 7

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Ring relations

There are 8 "obvious" relations:
   a_1_02, a_1_12, a_3_22, a_3_32, a_3_42, a_5_72, a_5_82, a_7_142

Apart from that, there are 80 minimal relations of maximal degree 13:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_1
  3. a_2_0·a_1_0
  4. a_2_1·a_1_1
  5. a_2_1·a_1_0
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9. a_1_1·a_3_2
  10. a_1_0·a_3_2
  11. a_1_1·a_3_4 − a_1_1·a_3_3
  12.  − a_1_1·a_3_3 + a_1_0·a_3_4 + a_1_0·a_3_3
  13. a_2_0·a_3_2
  14. a_2_1·a_3_2
  15. a_2_0·a_3_3
  16.  − a_2_1·a_3_3 + a_2_0·a_3_4
  17. a_2_1·a_3_4 + a_2_1·a_3_3
  18. a_4_4·a_1_1 + a_2_1·a_3_3
  19. a_4_4·a_1_0 − a_2_1·a_3_3
  20. b_4_5·a_1_1 + b_4_5·a_1_0
  21. a_3_2·a_3_4 − a_3_2·a_3_3
  22. a_3_3·a_3_4 − a_3_2·a_3_3 + c_2_2·a_1_0·a_3_4 + c_2_2·a_1_0·a_3_3
  23. a_2_0·a_4_4
  24. a_2_1·a_4_4
  25. a_2_0·b_4_5 − a_3_2·a_3_3
  26. a_2_1·b_4_5
  27.  − a_3_3·a_3_4 + a_1_1·a_5_7
  28. a_3_3·a_3_4 + a_1_0·a_5_7
  29. a_1_1·a_5_8 + a_1_0·a_5_8
  30. a_4_4·a_3_2
  31. a_4_4·a_3_4 − a_4_4·a_3_3
  32. b_4_5·a_3_4 − b_4_5·a_3_3 + b_4_5·a_3_2 + a_4_4·a_3_3 + c_2_2·b_4_5·a_1_0
       − a_2_0·c_2_2·a_3_4
  33. a_2_0·a_5_7 − a_2_0·c_2_2·a_3_4
  34. a_2_1·a_5_7
  35. a_4_4·a_3_3 + a_2_0·a_5_8
  36. a_2_1·a_5_8
  37. a_6_7·a_1_1 − a_4_4·a_3_3
  38. a_6_7·a_1_0 + a_4_4·a_3_3
  39. b_6_9·a_1_1 − b_4_5·a_3_4 + b_4_5·a_3_3 + b_4_5·a_3_2 + a_2_0·c_2_2·a_3_4
  40. b_6_9·a_1_0 + b_4_5·a_3_4 − b_4_5·a_3_3 − b_4_5·a_3_2 − a_2_0·c_2_2·a_3_4
  41. a_4_42
  42. a_3_3·a_5_7 + a_3_2·a_5_7 + b_4_5·a_1_0·a_3_3 − c_2_22·a_1_0·a_3_4
       − c_2_22·a_1_0·a_3_3
  43. a_3_4·a_5_7 − b_4_5·a_1_0·a_3_3
  44. a_3_3·a_5_7 + c_2_2·a_1_0·a_5_7 + c_2_22·a_1_0·a_3_4 + c_2_22·a_1_0·a_3_3
  45.  − a_4_4·b_4_5 + a_3_3·a_5_7 + a_3_2·a_5_8 + b_4_5·a_1_0·a_3_3 − c_2_22·a_1_0·a_3_4
       − c_2_22·a_1_0·a_3_3
  46. a_4_4·b_4_5 + a_3_4·a_5_8 − a_3_3·a_5_8 − a_3_3·a_5_7 − b_4_5·a_1_0·a_3_3
       + c_2_2·a_1_0·a_5_8 + c_2_22·a_1_0·a_3_4 + c_2_22·a_1_0·a_3_3
  47. a_2_0·a_6_7
  48. a_2_1·a_6_7
  49. a_2_0·b_6_9 − a_3_3·a_5_7 + b_4_5·a_1_0·a_3_3 + c_2_22·a_1_0·a_3_4
       + c_2_22·a_1_0·a_3_3
  50. a_2_1·b_6_9
  51. a_4_4·b_4_5 − a_3_4·a_5_8 + a_3_3·a_5_8 + a_1_1·a_7_14 − b_4_5·a_1_0·a_3_3
       + c_2_22·a_1_0·a_3_4 + c_2_22·a_1_0·a_3_3
  52.  − a_4_4·b_4_5 + a_3_4·a_5_8 − a_3_3·a_5_8 + a_1_0·a_7_14 + b_4_5·a_1_0·a_3_3
       − c_2_22·a_1_0·a_3_4 − c_2_22·a_1_0·a_3_3
  53. a_4_4·a_5_8 − a_4_4·a_5_7 − a_2_0·c_2_22·a_3_4
  54.  − a_4_4·a_5_7 + a_1_0·a_3_3·a_5_8 + a_2_0·c_2_2·a_5_8 − a_2_0·c_2_22·a_3_4
  55. a_6_7·a_3_3 − a_1_0·a_3_3·a_5_8 + a_2_0·c_2_22·a_3_4
  56. a_6_7·a_3_2 − a_1_0·a_3_3·a_5_8
  57. a_6_7·a_3_4 + a_4_4·a_5_7 − a_1_0·a_3_3·a_5_8 − a_2_0·c_2_22·a_3_4
  58. b_6_9·a_3_3 − b_4_5·a_5_7 + a_4_4·a_5_7 − a_1_0·a_3_3·a_5_8 + c_2_2·b_4_5·a_3_3
       − c_2_2·b_4_5·a_3_2 + c_2_22·b_4_5·a_1_0
  59. b_6_9·a_3_2 + b_4_52·a_1_0 + a_1_0·a_3_3·a_5_8 − c_2_2·b_4_5·a_3_2
  60. b_6_9·a_3_4 − b_4_5·a_5_7 − b_4_52·a_1_0 − a_4_4·a_5_7 + a_1_0·a_3_3·a_5_8
       + c_2_2·b_4_5·a_3_3 − c_2_2·b_4_5·a_3_2 − c_2_22·b_4_5·a_1_0
  61.  − a_4_4·a_5_7 + a_2_0·a_7_14 − a_1_0·a_3_3·a_5_8 + a_2_0·c_2_22·a_3_4
  62. a_2_1·a_7_14
  63. a_4_4·a_6_7
  64. b_4_5·a_6_7 + a_4_4·b_6_9 + a_5_7·a_5_8 − b_4_5·a_1_0·a_5_8 + c_2_2·a_3_3·a_5_8
       + c_2_2·b_4_5·a_1_0·a_3_3 − c_2_22·a_1_0·a_5_8 − c_2_22·a_1_0·a_5_7
       + c_2_23·a_1_0·a_3_4 + c_2_23·a_1_0·a_3_3
  65. b_4_5·a_6_7 + a_3_3·a_7_14 + b_4_5·a_1_0·a_5_8 − b_4_5·a_1_0·a_5_7 − c_6_11·a_1_0·a_3_4
       + c_6_11·a_1_0·a_3_3 − c_2_2·a_3_3·a_5_8 − c_2_2·b_4_5·a_1_0·a_3_3
       − c_2_22·a_1_0·a_5_7 − c_2_23·a_1_0·a_3_4 − c_2_23·a_1_0·a_3_3
  66. b_4_5·a_6_7 + a_5_7·a_5_8 + a_3_2·a_7_14 − b_4_5·a_1_0·a_5_8 + b_4_5·a_1_0·a_5_7
       + c_2_2·a_3_3·a_5_8 + c_2_2·b_4_5·a_1_0·a_3_3 − c_2_22·a_1_0·a_5_8
  67. b_4_5·a_6_7 + a_3_4·a_7_14 − b_4_5·a_1_0·a_5_7 + c_6_11·a_1_0·a_3_4 − c_6_11·a_1_0·a_3_3
       − c_2_2·a_3_3·a_5_8 + c_2_22·a_1_0·a_5_8 − c_2_22·a_1_0·a_5_7
  68.  − b_4_5·a_6_7 − a_5_7·a_5_8 − b_4_5·a_1_0·a_5_8 − b_4_5·a_1_0·a_5_7 − c_2_2·a_3_3·a_5_8
       + c_2_2·a_1_0·a_7_14 + c_2_2·b_4_5·a_1_0·a_3_3 − c_2_23·a_1_0·a_3_4
       − c_2_23·a_1_0·a_3_3
  69. a_6_7·a_5_8 + a_6_7·a_5_7 − a_2_0·c_6_11·a_3_4 + a_2_0·c_2_22·a_5_8
       − a_2_0·c_2_23·a_3_4
  70. b_6_9·a_5_7 − b_4_52·a_3_3 − c_2_2·b_4_52·a_1_0 + c_2_22·b_4_5·a_3_3
       − c_2_22·b_4_5·a_3_2 − c_2_23·b_4_5·a_1_0 + a_2_0·c_2_23·a_3_4
  71.  − a_6_7·a_5_7 + a_1_0·a_3_3·a_7_14 + a_2_0·c_2_22·a_5_8 + a_2_0·c_2_23·a_3_4
  72.  − a_6_7·a_5_7 + a_4_4·a_7_14 − c_2_2·a_1_0·a_3_3·a_5_8 − a_2_0·c_2_22·a_5_8
       + a_2_0·c_2_23·a_3_4
  73.  − b_6_9·a_5_8 + b_4_5·a_7_14 + b_4_52·a_3_2 + a_6_7·a_5_7 + c_2_2·b_4_5·a_5_7
       + c_2_2·b_4_52·a_1_0 − a_2_0·c_6_11·a_3_4 − c_2_2·a_1_0·a_3_3·a_5_8
       − c_2_22·b_4_5·a_3_3 − c_2_22·b_4_5·a_3_2 + a_2_0·c_2_22·a_5_8
  74. a_6_72
  75. b_6_92 − b_4_53 + b_4_5·a_3_3·a_5_8 + b_4_52·a_1_0·a_3_3 + c_2_2·b_4_5·b_6_9
       − c_2_2·b_4_5·a_1_0·a_5_7 + c_2_22·b_4_52 + c_2_22·b_4_5·a_1_0·a_3_3
  76. a_6_7·b_6_9 + b_4_5·a_3_3·a_5_8 + b_4_5·a_1_0·a_7_14 + b_4_52·a_1_0·a_3_3
       + c_2_2·a_3_3·a_7_14 + c_2_2·b_4_5·a_1_0·a_5_7 − c_2_2·c_6_11·a_1_0·a_3_4
       + c_2_2·c_6_11·a_1_0·a_3_3 − c_2_22·a_3_3·a_5_8 − c_2_23·a_1_0·a_5_7
       − c_2_24·a_1_0·a_3_4 − c_2_24·a_1_0·a_3_3
  77.  − a_6_7·b_6_9 + a_5_8·a_7_14 − a_5_7·a_7_14 + b_4_52·a_1_0·a_3_3 + c_2_2·a_3_3·a_7_14
       − c_2_2·b_4_5·a_1_0·a_5_8 + c_2_2·b_4_5·a_1_0·a_5_7 − c_2_2·c_6_11·a_1_0·a_3_4
       + c_2_2·c_6_11·a_1_0·a_3_3 + c_2_22·b_4_5·a_1_0·a_3_3
  78. a_5_7·a_7_14 − b_4_5·a_3_3·a_5_8 + b_4_52·a_1_0·a_3_3 − c_2_2·b_4_5·a_1_0·a_5_8
       + c_2_2·b_4_5·a_1_0·a_5_7 + c_2_22·a_3_3·a_5_8 + c_2_22·a_1_0·a_7_14
       + c_2_22·b_4_5·a_1_0·a_3_3 + c_2_23·a_1_0·a_5_8 + c_2_23·a_1_0·a_5_7
       − c_2_24·a_1_0·a_3_4 − c_2_24·a_1_0·a_3_3
  79. b_6_9·a_7_14 − b_4_52·a_5_8 − b_4_53·a_1_0 + b_4_5·a_1_0·a_3_3·a_5_8
       + c_2_2·b_4_5·a_7_14 + c_2_2·b_4_52·a_3_3 + c_2_2·b_4_52·a_3_2
       − c_2_2·a_1_0·a_3_3·a_7_14 + c_2_22·b_4_5·a_5_8 + c_2_22·b_4_52·a_1_0
       + a_2_0·c_2_2·c_6_11·a_3_4 − c_2_22·a_1_0·a_3_3·a_5_8 − c_2_23·b_4_5·a_3_3
       + c_2_23·b_4_5·a_3_2 − a_2_0·c_2_23·a_5_8 − c_2_24·b_4_5·a_1_0
  80. a_6_7·a_7_14 − b_4_5·a_1_0·a_3_3·a_5_8 − c_2_2·a_1_0·a_3_3·a_7_14
       − a_2_0·c_2_2·c_6_11·a_3_4 − c_2_22·a_1_0·a_3_3·a_5_8 − a_2_0·c_2_24·a_3_4

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

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_2_2, a Duflot regular element of degree 2
    2. c_6_11, a Duflot regular element of degree 6
    3. b_4_5, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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

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. c_2_2 → c_2_1, an element of degree 2
  6. a_3_2 → 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. a_4_4 → 0, an element of degree 4
  10. b_4_5 → 0, an element of degree 4
  11. a_5_7 → 0, an element of degree 5
  12. a_5_8 → 0, an element of degree 5
  13. a_6_7 → 0, an element of degree 6
  14. b_6_9 → 0, an element of degree 6
  15. c_6_11 →  − c_2_23, an element of degree 6
  16. a_7_14 → 0, an element of degree 7

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. c_2_2 → c_2_3, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_3_3 → c_2_5·a_1_2, an element of degree 3
  8. a_3_4 → c_2_5·a_1_2, an element of degree 3
  9. a_4_4 → 0, an element of degree 4
  10. b_4_5 → c_2_52, an element of degree 4
  11. a_5_7 →  − c_2_52·a_1_2 − c_2_3·c_2_5·a_1_2, an element of degree 5
  12. a_5_8 → c_2_52·a_1_2 + c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
  13. a_6_7 →  − c_2_52·a_1_1·a_1_2, an element of degree 6
  14. b_6_9 → c_2_52·a_1_1·a_1_2 − c_2_53 + c_2_3·c_2_52, an element of degree 6
  15. c_6_11 → c_2_52·a_1_1·a_1_2 + c_2_4·c_2_52 − c_2_43 + c_2_3·c_2_52, an element of degree 6
  16. a_7_14 →  − c_2_53·a_1_2 − c_2_53·a_1_1 + c_2_4·c_2_52·a_1_2 − c_2_3·c_2_52·a_1_2
       + c_2_3·c_2_52·a_1_1 − c_2_3·c_2_4·c_2_5·a_1_2 − c_2_32·c_2_5·a_1_2, 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