Mod-3-Cohomology of group number 17 of order 108

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General information on the group

  • The group order factors as 22 · 33.
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

The computation was based on 3 stability conditions for H*(E27; GF(3)).

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
    1  −  2·t  +  3·t2  −  t3  +  2·t4  −  2·t5  +  2·t6  −  2·t7  +  2·t8  −  t9  +  3·t10  −  2·t11  +  t12
    ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Ring generators

The cohomology ring has 16 minimal generators of maximal degree 12:

  1. a_3_2, a nilpotent element of degree 3
  2. a_3_1, a nilpotent element of degree 3
  3. a_3_0, a nilpotent element of degree 3
  4. a_4_3, a nilpotent element of degree 4
  5. b_4_2, an element of degree 4
  6. b_4_1, an element of degree 4
  7. b_4_0, an element of degree 4
  8. a_5_0, a nilpotent element of degree 5
  9. a_9_1, a nilpotent element of degree 9
  10. a_10_2, a nilpotent element of degree 10
  11. b_10_1, an element of degree 10
  12. a_10_0, a nilpotent element of degree 10
  13. a_11_5, a nilpotent element of degree 11
  14. a_11_4, a nilpotent element of degree 11
  15. a_11_1, a nilpotent element of degree 11
  16. c_12_4, a Duflot element of degree 12

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Ring relations

There are 8 "obvious" relations:
   a_3_02, a_3_12, a_3_22, a_5_02, a_9_12, a_11_12, a_11_42, a_11_52

Apart from that, there are 96 minimal relations of maximal degree 22:

  1. a_3_0·a_3_1
  2. a_3_0·a_3_2
  3. a_3_1·a_3_2
  4. a_4_3·a_3_0
  5. a_4_3·a_3_1
  6. a_4_3·a_3_2
  7. b_4_0·a_3_1
  8. b_4_1·a_3_0 + b_4_0·a_3_2
  9. b_4_1·a_3_1
  10. b_4_1·a_3_2 + b_4_0·a_3_2
  11. b_4_2·a_3_0 − b_4_0·a_3_2
  12. b_4_2·a_3_2 − b_4_2·a_3_1 − b_4_0·a_3_2
  13. a_4_32
  14. a_3_1·a_5_0
  15. a_3_2·a_5_0
  16. a_4_3·b_4_0 − a_3_0·a_5_0
  17. a_4_3·b_4_1
  18. a_4_3·b_4_2
  19. b_4_0·b_4_2 + b_4_0·b_4_1
  20. b_4_12 + b_4_0·b_4_1
  21. b_4_1·b_4_2 − b_4_0·b_4_1
  22. a_4_3·a_5_0
  23. b_4_1·a_5_0
  24. b_4_2·a_5_0
  25. a_3_0·a_9_1
  26. a_3_1·a_9_1
  27. a_3_2·a_9_1
  28. a_4_3·a_9_1
  29. a_10_0·a_3_0
  30. a_10_0·a_3_1
  31. a_10_0·a_3_2
  32. a_10_2·a_3_0
  33. a_10_2·a_3_1
  34. a_10_2·a_3_2
  35. b_4_1·a_9_1
  36. b_4_2·a_9_1
  37. b_10_1·a_3_0 − b_4_0·a_9_1
  38. b_10_1·a_3_1
  39. b_10_1·a_3_2
  40. a_4_3·a_10_0
  41. a_4_3·a_10_2
  42. a_3_0·a_11_5
  43. a_3_1·a_11_4 − a_3_1·a_11_1
  44. a_3_1·a_11_5 + a_3_1·a_11_1
  45. a_3_2·a_11_1 − a_3_1·a_11_1 − a_3_0·a_11_1
  46. a_3_2·a_11_4 − a_3_1·a_11_1
  47. a_3_2·a_11_5 + a_3_1·a_11_1
  48. a_5_0·a_9_1 + a_3_0·a_11_4
  49. a_4_3·b_10_1 − a_3_0·a_11_4
  50. b_4_0·a_10_0 − a_3_0·a_11_1
  51. b_4_0·a_10_2 − a_3_0·a_11_4
  52. b_4_1·a_10_0 + a_3_0·a_11_1
  53. b_4_1·a_10_2
  54. b_4_2·a_10_0 − a_3_1·a_11_1 − a_3_0·a_11_1
  55. b_4_2·a_10_2 − a_3_1·a_11_1
  56. b_4_1·b_10_1
  57. b_4_2·b_10_1
  58. a_4_3·a_11_1
  59. a_4_3·a_11_4
  60. a_4_3·a_11_5
  61. a_10_0·a_5_0
  62. a_10_2·a_5_0
  63. b_4_0·a_11_5
  64. b_4_1·a_11_1 + b_4_0·a_11_1
  65. b_4_1·a_11_4
  66. b_4_1·a_11_5
  67. b_4_2·a_11_4 − b_4_2·a_11_1 + b_4_0·a_11_1
  68. b_4_2·a_11_5 + b_4_2·a_11_1 − b_4_0·a_11_1
  69. b_10_1·a_5_0 − b_4_0·a_11_4
  70. a_5_0·a_11_1
  71. a_5_0·a_11_4
  72. a_5_0·a_11_5
  73. a_10_0·a_9_1
  74. a_10_2·a_9_1
  75. b_10_1·a_9_1 + b_4_0·c_12_4·a_3_2 − b_4_0·c_12_4·a_3_0
  76. a_10_02
  77. a_10_0·a_10_2
  78. a_10_22
  79. a_9_1·a_11_1
  80. a_9_1·a_11_4 − c_12_4·a_3_0·a_5_0
  81. a_9_1·a_11_5
  82. a_10_0·b_10_1
  83. a_10_2·b_10_1 − c_12_4·a_3_0·a_5_0
  84. b_10_12 − b_4_0·b_4_1·c_12_4 − b_4_02·c_12_4
  85. a_10_0·a_11_1
  86. a_10_0·a_11_4
  87. a_10_0·a_11_5
  88. a_10_2·a_11_1
  89. a_10_2·a_11_4
  90. a_10_2·a_11_5
  91. b_10_1·a_11_1
  92. b_10_1·a_11_4 − b_4_0·c_12_4·a_5_0
  93. b_10_1·a_11_5
  94. a_11_1·a_11_4
  95. a_11_1·a_11_5
  96. a_11_4·a_11_5

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 22 using the Hilbert-Poincaré criterion.
  • 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_12_4, an element of degree 12
    2. b_4_2 + b_4_0, an element of degree 4
  • A Duflot regular sequence is given by c_12_4.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 14].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(E27; GF(3))

  1. a_3_2 → b_2_0·a_1_1 − b_2_0·a_1_0
  2. a_3_1 → b_2_1·a_1_1 − b_2_0·a_1_0
  3. a_3_0 → b_2_3·a_1_1
  4. a_4_3 → a_1_1·a_3_5
  5. b_4_2 → b_2_0·b_2_1 − b_2_02 + a_1_0·a_3_4
  6. b_4_1 → b_2_0·b_2_2 − a_1_0·a_3_5
  7. b_4_0 → b_2_32
  8. a_5_0 → b_2_3·a_3_5 + b_2_0·b_2_1·a_1_1 − b_2_02·a_1_1
  9. a_9_1 → b_2_3·c_6_8·a_1_1 + b_2_0·c_6_8·a_1_1
  10. a_10_2 → c_6_8·a_1_1·a_3_5 + c_6_8·a_1_0·a_3_5
  11. b_10_1 → b_2_32·c_6_8 + b_2_0·b_2_1·c_6_8 + c_6_8·a_1_0·a_3_4
  12. a_10_0 → b_2_03·a_1_1·a_3_5 + b_2_03·a_1_0·a_3_5 − b_2_03·a_1_0·a_3_4 − c_6_8·a_1_0·a_3_5
       − c_6_8·a_1_0·a_3_4
  13. a_11_5 → b_2_1·c_6_8·a_3_5 − b_2_0·c_6_8·a_3_5 − b_2_0·b_2_1·c_6_8·a_1_1 + b_2_02·c_6_8·a_1_1
  14. a_11_4 → b_2_3·c_6_8·a_3_5 + b_2_0·c_6_8·a_3_5
  15. a_11_1 → b_2_03·b_2_1·a_3_5 + b_2_04·a_3_5 − b_2_04·a_3_4 − b_2_0·c_6_8·a_3_5
       − b_2_0·c_6_8·a_3_4
  16. c_12_4 → b_2_04·a_1_1·a_3_5 + b_2_04·a_1_0·a_3_5 − b_2_04·a_1_0·a_3_4 + b_2_02·b_2_2·c_6_8
       − b_2_0·c_6_8·a_1_1·a_3_5 − b_2_0·c_6_8·a_1_0·a_3_5 − b_2_0·c_6_8·a_1_0·a_3_4
       + c_6_82

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_3_2 → 0, an element of degree 3
  2. a_3_1 → 0, an element of degree 3
  3. a_3_0 → 0, an element of degree 3
  4. a_4_3 → 0, an element of degree 4
  5. b_4_2 → 0, an element of degree 4
  6. b_4_1 → 0, an element of degree 4
  7. b_4_0 → 0, an element of degree 4
  8. a_5_0 → 0, an element of degree 5
  9. a_9_1 → 0, an element of degree 9
  10. a_10_2 → 0, an element of degree 10
  11. b_10_1 → 0, an element of degree 10
  12. a_10_0 → 0, an element of degree 10
  13. a_11_5 → 0, an element of degree 11
  14. a_11_4 → 0, an element of degree 11
  15. a_11_1 → 0, an element of degree 11
  16. c_12_4 → c_2_06, an element of degree 12

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

  1. a_3_2 →  − c_2_2·a_1_1, an element of degree 3
  2. a_3_1 →  − c_2_2·a_1_1, an element of degree 3
  3. a_3_0 → 0, an element of degree 3
  4. a_4_3 → 0, an element of degree 4
  5. b_4_2 →  − c_2_22, an element of degree 4
  6. b_4_1 → 0, an element of degree 4
  7. b_4_0 → 0, an element of degree 4
  8. a_5_0 → 0, an element of degree 5
  9. a_9_1 → 0, an element of degree 9
  10. a_10_2 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  11. b_10_1 → 0, an element of degree 10
  12. a_10_0 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  13. a_11_5 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  14. a_11_4 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  15. a_11_1 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  16. c_12_4 → c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12

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

  1. a_3_2 → 0, an element of degree 3
  2. a_3_1 → 0, an element of degree 3
  3. a_3_0 → c_2_2·a_1_1, an element of degree 3
  4. a_4_3 → c_2_2·a_1_0·a_1_1, an element of degree 4
  5. b_4_2 → 0, an element of degree 4
  6. b_4_1 → 0, an element of degree 4
  7. b_4_0 → c_2_22, an element of degree 4
  8. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  9. a_9_1 → c_2_1·c_2_23·a_1_1 − c_2_13·c_2_2·a_1_1, an element of degree 9
  10. a_10_2 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  11. b_10_1 → c_2_1·c_2_24 − c_2_13·c_2_22, an element of degree 10
  12. a_10_0 → 0, an element of degree 10
  13. a_11_5 → 0, an element of degree 11
  14. a_11_4 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  15. a_11_1 → 0, an element of degree 11
  16. c_12_4 → c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12

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

  1. a_3_2 → 0, an element of degree 3
  2. a_3_1 → 0, an element of degree 3
  3. a_3_0 → c_2_2·a_1_1, an element of degree 3
  4. a_4_3 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
  5. b_4_2 → 0, an element of degree 4
  6. b_4_1 → 0, an element of degree 4
  7. b_4_0 → c_2_22, an element of degree 4
  8. a_5_0 → c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  9. a_9_1 →  − c_2_1·c_2_23·a_1_1 + c_2_13·c_2_2·a_1_1, an element of degree 9
  10. a_10_2 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  11. b_10_1 →  − c_2_1·c_2_24 + c_2_13·c_2_22, an element of degree 10
  12. a_10_0 → 0, an element of degree 10
  13. a_11_5 → 0, an element of degree 11
  14. a_11_4 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  15. a_11_1 → 0, an element of degree 11
  16. c_12_4 → c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12

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

  1. a_3_2 → c_2_2·a_1_1, an element of degree 3
  2. a_3_1 → 0, an element of degree 3
  3. a_3_0 → c_2_2·a_1_1, an element of degree 3
  4. a_4_3 → 0, an element of degree 4
  5. b_4_2 → c_2_22, an element of degree 4
  6. b_4_1 →  − c_2_22, an element of degree 4
  7. b_4_0 → c_2_22, an element of degree 4
  8. a_5_0 → 0, an element of degree 5
  9. a_9_1 → 0, an element of degree 9
  10. a_10_2 → 0, an element of degree 10
  11. b_10_1 → 0, an element of degree 10
  12. a_10_0 →  − c_2_1·c_2_23·a_1_0·a_1_1 + c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  13. a_11_5 → 0, an element of degree 11
  14. a_11_4 → 0, an element of degree 11
  15. a_11_1 →  − c_2_25·a_1_1 + c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  16. c_12_4 →  − c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010