Mod-3-Cohomology of AlternatingGroup(9), a group of order 181440

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General information on the group

  • AlternatingGroup(9) is a group of order 181440.
  • The group order factors as 26 · 34 · 5 · 7.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9),(7,8,9)]).
  • It is non-abelian.
  • It has 3-Rank 3.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

The computation was based on 3 stability conditions for H*(SmallGroup(162,19); GF(3)).

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1)·(1  −  2·t  +  2·t2  −  t3  +  2·t4  −  4·t5  +  3·t6  +  2·t8  −  4·t9  +  2·t10  −  t11  +  2·t12  −  2·t13  +  t14)
    ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4) · (1  +  t4)
  • The a-invariants are -∞,-∞,-9,-3. 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, -3, -3].

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Ring generators

The cohomology ring has 16 minimal generators of maximal degree 18:

  1. a_3_1, a nilpotent element of degree 3
  2. a_3_0, a nilpotent element of degree 3
  3. a_4_1, a nilpotent element of degree 4
  4. b_4_0, an element of degree 4
  5. a_7_1, a nilpotent element of degree 7
  6. a_7_0, a nilpotent element of degree 7
  7. a_8_3, a nilpotent element of degree 8
  8. b_8_1, an element of degree 8
  9. b_8_0, an element of degree 8
  10. a_9_0, a nilpotent element of degree 9
  11. a_11_0, a nilpotent element of degree 11
  12. a_12_3, a nilpotent element of degree 12
  13. c_12_0, a Duflot element of degree 12
  14. a_13_0, a nilpotent element of degree 13
  15. a_17_1, a nilpotent element of degree 17
  16. b_18_0, an element of degree 18

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Ring relations

There are 8 "obvious" relations:
   a_3_02, a_3_12, a_7_02, a_7_12, a_9_02, a_11_02, a_13_02, a_17_12

Apart from that, there are 80 minimal relations of maximal degree 36:

  1. a_3_0·a_3_1
  2. a_4_1·a_3_1
  3. b_4_0·a_3_1 − a_4_1·a_3_0
  4. a_4_12
  5. a_3_0·a_7_0
  6. a_3_1·a_7_1
  7. a_4_1·a_7_0
  8. a_8_3·a_3_0 + a_4_1·a_7_1
  9. a_8_3·a_3_1
  10. b_4_0·a_7_0
  11. b_8_0·a_3_0
  12. b_8_1·a_3_1 − a_4_1·a_7_1
  13. a_4_1·a_8_3
  14. a_3_1·a_9_0
  15. a_4_1·b_8_0
  16. b_4_0·a_8_3 + a_4_1·b_8_1 − a_3_0·a_9_0
  17. b_4_0·b_8_0
  18. a_4_1·a_9_0
  19. a_3_1·a_11_0
  20. a_7_0·a_7_1
  21. a_4_1·a_11_0
  22. a_8_3·a_7_0
  23. a_8_3·a_7_1 − a_4_1·b_4_0·a_7_1
  24. a_12_3·a_3_0
  25. a_12_3·a_3_1
  26. b_8_0·a_7_1
  27. b_8_1·a_7_0
  28. a_4_1·a_12_3
  29. a_8_32
  30. a_3_1·a_13_0
  31. a_7_0·a_9_0
  32. b_4_0·a_12_3 − a_3_0·a_13_0
  33. a_8_3·b_8_0
  34. a_8_3·b_8_1 − a_4_1·b_4_0·b_8_1 − a_7_1·a_9_0
  35. b_8_0·b_8_1
  36. a_4_1·a_13_0
  37. a_8_3·a_9_0
  38. b_8_0·a_9_0
  39. a_7_0·a_11_0
  40. a_8_3·a_11_0 − a_3_0·a_7_1·a_9_0 + a_4_1·c_12_0·a_3_0
  41. a_12_3·a_7_0
  42. a_12_3·a_7_1 − a_3_0·a_7_1·a_9_0 + a_4_1·c_12_0·a_3_0
  43. b_8_0·a_11_0
  44. a_8_3·a_12_3
  45. a_3_1·a_17_1
  46. a_7_0·a_13_0
  47. a_7_1·a_13_0 + a_3_0·a_17_1 − b_8_1·a_3_0·a_9_0 + b_4_0·a_7_1·a_9_0 − a_4_1·b_4_0·c_12_0
  48. a_9_0·a_11_0 + b_8_1·a_3_0·a_9_0 − b_4_0·a_7_1·a_9_0 + a_4_1·b_4_0·c_12_0
  49. b_8_0·a_12_3
  50. b_8_1·a_12_3 + a_3_0·a_17_1
  51. a_4_1·a_17_1 − a_3_0·a_7_1·a_11_0
  52. a_8_3·a_13_0 + a_3_0·a_7_1·a_11_0
  53. a_12_3·a_9_0 − a_3_0·a_7_1·a_11_0
  54. b_8_0·a_13_0
  55. b_18_0·a_3_0 − b_8_1·a_13_0 − b_4_0·a_17_1
  56. b_18_0·a_3_1 − a_3_0·a_7_1·a_11_0
  57. a_9_0·a_13_0 − b_8_1·a_3_0·a_11_0 + b_4_0·a_7_1·a_11_0
  58. a_4_1·b_18_0 + b_8_1·a_3_0·a_11_0 − b_4_0·a_7_1·a_11_0
  59. a_12_3·a_11_0 + b_4_0·a_3_0·a_7_1·a_9_0 − a_4_1·c_12_0·a_7_1
  60. a_12_32
  61. a_7_0·a_17_1
  62. a_7_1·a_17_1 − b_8_1·a_7_1·a_9_0 + b_4_0·a_3_0·a_17_1 − b_4_0·b_8_1·a_3_0·a_9_0
       + a_4_1·b_8_1·c_12_0 − c_12_0·a_3_0·a_9_0
  63. a_11_0·a_13_0 + b_4_0·b_8_1·a_3_0·a_9_0 − b_4_02·a_7_1·a_9_0 + a_4_1·b_8_1·c_12_0
  64. a_8_3·a_17_1
  65. a_12_3·a_13_0
  66. b_8_0·a_17_1
  67. b_18_0·a_7_0
  68. b_18_0·a_7_1 − b_8_1·a_17_1 + b_8_12·a_9_0 + b_4_0·b_8_1·a_13_0 + b_4_02·b_8_1·a_9_0
       + b_4_0·c_12_0·a_9_0
  69. a_9_0·a_17_1 − b_8_1·a_7_1·a_11_0 − b_8_12·a_3_0·a_7_1 − b_4_0·b_8_1·a_3_0·a_11_0
       − b_4_02·b_8_1·a_3_0·a_7_1 − b_4_0·c_12_0·a_3_0·a_7_1
  70. a_8_3·b_18_0 + b_8_1·a_7_1·a_11_0 + b_8_12·a_3_0·a_7_1 + b_4_0·b_8_1·a_3_0·a_11_0
       + b_4_02·b_8_1·a_3_0·a_7_1 + b_4_0·c_12_0·a_3_0·a_7_1
  71. b_8_0·b_18_0
  72. b_18_0·a_9_0 + b_8_12·a_11_0 − b_8_13·a_3_0 + b_4_0·b_8_12·a_7_1
       + b_4_02·b_8_1·a_11_0 − b_4_02·b_8_12·a_3_0 + b_4_03·b_8_1·a_7_1
       − b_4_0·b_8_1·c_12_0·a_3_0 + b_4_02·c_12_0·a_7_1
  73. a_11_0·a_17_1 − b_8_1·a_3_0·a_17_1 + b_4_0·b_8_1·a_7_1·a_9_0 − b_4_02·a_3_0·a_17_1
       + b_4_02·b_8_1·a_3_0·a_9_0 + a_4_1·b_4_0·b_8_1·c_12_0 + c_12_0·a_7_1·a_9_0
       + c_12_0·a_3_0·a_13_0
  74. a_12_3·a_17_1
  75. b_18_0·a_11_0 − b_8_12·a_13_0 − b_4_0·b_8_12·a_9_0 − b_4_02·b_8_1·a_13_0
       − b_4_03·b_8_1·a_9_0 − b_8_1·c_12_0·a_9_0 − b_4_0·c_12_0·a_13_0
  76. a_13_0·a_17_1 + b_8_12·a_3_0·a_11_0 + b_4_0·b_8_12·a_3_0·a_7_1
       + b_4_02·b_8_1·a_3_0·a_11_0 + b_4_03·b_8_1·a_3_0·a_7_1 + b_8_1·c_12_0·a_3_0·a_7_1
       + b_4_0·c_12_0·a_3_0·a_11_0
  77. a_12_3·b_18_0 − b_8_12·a_3_0·a_11_0 − b_4_0·b_8_12·a_3_0·a_7_1
       − b_4_02·b_8_1·a_3_0·a_11_0 − b_4_03·b_8_1·a_3_0·a_7_1 − b_8_1·c_12_0·a_3_0·a_7_1
       − b_4_0·c_12_0·a_3_0·a_11_0
  78. b_18_0·a_13_0 − b_4_0·b_8_12·a_11_0 + b_4_0·b_8_13·a_3_0 − b_4_02·b_8_12·a_7_1
       − b_4_03·b_8_1·a_11_0 + b_4_03·b_8_12·a_3_0 − b_4_04·b_8_1·a_7_1
       + b_8_12·c_12_0·a_3_0 − b_4_0·b_8_1·c_12_0·a_7_1 − b_4_02·c_12_0·a_11_0
  79. b_18_0·a_17_1 + b_8_13·a_11_0 − b_8_14·a_3_0 + b_4_0·b_8_13·a_7_1
       + b_4_02·b_8_12·a_11_0 − b_4_02·b_8_13·a_3_0 + b_4_03·b_8_12·a_7_1
       + b_8_12·c_12_0·a_7_1 + b_4_0·b_8_1·c_12_0·a_11_0 − b_4_0·b_8_12·c_12_0·a_3_0
       + b_4_03·b_8_1·c_12_0·a_3_0 − b_4_02·c_12_02·a_3_0
  80. b_18_02 + b_8_13·c_12_0 − b_4_02·b_8_12·c_12_0 + b_4_04·b_8_1·c_12_0
       − b_4_03·c_12_02

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 36 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.  − b_4_03·b_8_13 + b_4_09 + b_8_13·c_12_0 + b_4_02·b_8_12·c_12_0 + c_12_03, an element of degree 36
    2.  − b_8_16 + b_8_06 − b_4_06·b_8_13 − b_4_0·b_8_14·c_12_0 + b_4_05·b_8_12·c_12_0
         − b_8_13·c_12_02 + b_4_02·b_8_12·c_12_02 + b_4_03·c_12_03, an element of degree 48
    3. b_8_1, an element of degree 8
  • A Duflot regular sequence is given by c_12_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 75, 89].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. b_4_03 + c_12_0, an element of degree 12
    2. b_8_0 + b_4_02, an element of degree 8
    3. b_8_1, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(162,19); GF(3))

  1. a_3_1 → a_3_1
  2. a_3_0 → a_3_2 − a_3_0
  3. a_4_1 → a_4_3
  4. b_4_0 → b_4_1 − b_4_0
  5. a_7_1 → a_7_9 + a_7_1
  6. a_7_0 → a_7_10 + b_4_2·a_3_1
  7. a_8_3 → a_8_11 + a_3_2·a_5_0 − a_3_0·a_5_0
  8. b_8_1 → b_8_9
  9. b_8_0 → b_8_10
  10. a_9_0 → a_9_6 + b_6_0·a_3_0 + b_4_0·a_5_1 − b_4_0·a_5_0
  11. a_11_0 → a_11_13 − b_4_1·a_7_1 − b_4_0·b_4_1·a_3_0 + b_4_02·a_3_2
  12. a_12_3 → a_12_16 + b_4_1·a_3_0·a_5_0 + b_4_0·a_3_2·a_5_1 + b_4_0·a_3_0·a_5_1
  13. c_12_0 → b_4_23 + b_4_0·b_8_9 + b_4_02·b_4_1 + c_12_20
  14. a_13_0 → a_13_10 − b_4_1·b_6_0·a_3_0 + b_4_0·b_4_1·a_5_1 + b_4_0·b_4_1·a_5_0 + b_4_02·a_5_1
  15. a_17_1 → b_4_1·b_6_0·a_7_9 − b_4_0·b_4_1·a_9_6 + b_4_0·b_4_1·b_6_0·a_3_0 − b_4_0·b_4_12·a_5_0
       + b_4_02·b_4_1·a_5_1 − c_12_20·a_5_0
  16. b_18_0 → b_4_1·b_6_0·b_8_9 − b_4_0·b_4_12·b_6_0 + b_4_02·b_4_1·b_6_0 − b_6_0·c_12_20

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

  1. a_3_1 → 0, an element of degree 3
  2. a_3_0 → 0, an element of degree 3
  3. a_4_1 → 0, an element of degree 4
  4. b_4_0 → 0, an element of degree 4
  5. a_7_1 → 0, an element of degree 7
  6. a_7_0 → 0, an element of degree 7
  7. a_8_3 → 0, an element of degree 8
  8. b_8_1 → 0, an element of degree 8
  9. b_8_0 → 0, an element of degree 8
  10. a_9_0 → 0, an element of degree 9
  11. a_11_0 → 0, an element of degree 11
  12. a_12_3 → 0, an element of degree 12
  13. c_12_0 → c_2_06, an element of degree 12
  14. a_13_0 → 0, an element of degree 13
  15. a_17_1 → 0, an element of degree 17
  16. b_18_0 → 0, an element of degree 18

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

  1. a_3_1 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  2. a_3_0 → 0, an element of degree 3
  3. a_4_1 → 0, an element of degree 4
  4. b_4_0 → 0, an element of degree 4
  5. a_7_1 → 0, an element of degree 7
  6. a_7_0 →  − c_2_23·a_1_0 + c_2_13·a_1_1, an element of degree 7
  7. a_8_3 → 0, an element of degree 8
  8. b_8_1 → 0, an element of degree 8
  9. b_8_0 →  − c_2_1·c_2_23 + c_2_13·c_2_2, an element of degree 8
  10. a_9_0 → 0, an element of degree 9
  11. a_11_0 → 0, an element of degree 11
  12. a_12_3 → 0, an element of degree 12
  13. c_12_0 → c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
  14. a_13_0 → 0, an element of degree 13
  15. a_17_1 → 0, an element of degree 17
  16. b_18_0 → 0, an element of degree 18

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

  1. a_3_1 →  − a_1_0·a_1_1·a_1_2, an element of degree 3
  2. a_3_0 → c_2_5·a_1_2 + c_2_5·a_1_1 + c_2_4·a_1_2 + c_2_4·a_1_1 − c_2_4·a_1_0 − c_2_3·a_1_1, an element of degree 3
  3. a_4_1 →  − c_2_5·a_1_0·a_1_1 + c_2_4·a_1_0·a_1_2 − c_2_3·a_1_1·a_1_2, an element of degree 4
  4. b_4_0 → c_2_52 − c_2_4·c_2_5 + c_2_42 + c_2_3·c_2_4, an element of degree 4
  5. a_7_1 →  − c_2_4·c_2_52·a_1_0 + c_2_42·c_2_5·a_1_0 − c_2_3·c_2_52·a_1_1
       + c_2_3·c_2_4·c_2_5·a_1_2 − c_2_3·c_2_4·c_2_5·a_1_1 + c_2_3·c_2_42·a_1_2
       − c_2_3·c_2_42·a_1_0 − c_2_32·c_2_4·a_1_1 + c_2_33·a_1_1, an element of degree 7
  6. a_7_0 → 0, an element of degree 7
  7. a_8_3 →  − c_2_4·c_2_52·a_1_0·a_1_2 − c_2_4·c_2_52·a_1_0·a_1_1 + c_2_42·c_2_5·a_1_0·a_1_2
       + c_2_42·c_2_5·a_1_0·a_1_1 + c_2_3·c_2_52·a_1_1·a_1_2
       − c_2_3·c_2_4·c_2_5·a_1_1·a_1_2 + c_2_3·c_2_4·c_2_5·a_1_0·a_1_1
       + c_2_3·c_2_42·a_1_1·a_1_2 − c_2_3·c_2_42·a_1_0·a_1_2 + c_2_32·c_2_4·a_1_1·a_1_2
       − c_2_33·a_1_1·a_1_2, an element of degree 8
  8. b_8_1 →  − c_2_3·c_2_4·c_2_52 + c_2_3·c_2_42·c_2_5 + c_2_32·c_2_42 + c_2_33·c_2_4, an element of degree 8
  9. b_8_0 → 0, an element of degree 8
  10. a_9_0 → c_2_4·c_2_53·a_1_0 − c_2_43·c_2_5·a_1_0 − c_2_3·c_2_53·a_1_1 + c_2_3·c_2_43·a_1_2
       + c_2_33·c_2_5·a_1_1 − c_2_33·c_2_4·a_1_2, an element of degree 9
  11. a_11_0 → c_2_4·c_2_54·a_1_1 − c_2_42·c_2_53·a_1_2 − c_2_43·c_2_52·a_1_1
       − c_2_43·c_2_52·a_1_0 + c_2_44·c_2_5·a_1_2 + c_2_44·c_2_5·a_1_0
       + c_2_3·c_2_54·a_1_0 + c_2_3·c_2_4·c_2_53·a_1_0 + c_2_3·c_2_42·c_2_52·a_1_0
       + c_2_3·c_2_43·c_2_5·a_1_2 + c_2_3·c_2_43·c_2_5·a_1_1 + c_2_3·c_2_44·a_1_2
       − c_2_3·c_2_44·a_1_0 − c_2_32·c_2_53·a_1_2 − c_2_32·c_2_53·a_1_1
       + c_2_32·c_2_4·c_2_52·a_1_1 + c_2_32·c_2_42·c_2_5·a_1_2 + c_2_32·c_2_43·a_1_1
       − c_2_33·c_2_52·a_1_1 − c_2_33·c_2_52·a_1_0 + c_2_33·c_2_4·c_2_5·a_1_2
       − c_2_33·c_2_4·c_2_5·a_1_1 + c_2_33·c_2_4·c_2_5·a_1_0 + c_2_33·c_2_42·a_1_2
       − c_2_33·c_2_42·a_1_0 + c_2_34·c_2_5·a_1_2 + c_2_34·c_2_5·a_1_1
       + c_2_34·c_2_4·a_1_2 + c_2_34·c_2_4·a_1_1 − c_2_34·c_2_4·a_1_0 + c_2_35·a_1_1, an element of degree 11
  12. a_12_3 →  − c_2_4·c_2_54·a_1_1·a_1_2 − c_2_42·c_2_53·a_1_1·a_1_2
       − c_2_42·c_2_53·a_1_0·a_1_1 + c_2_43·c_2_52·a_1_1·a_1_2
       − c_2_43·c_2_52·a_1_0·a_1_2 − c_2_43·c_2_52·a_1_0·a_1_1
       + c_2_44·c_2_5·a_1_1·a_1_2 + c_2_44·c_2_5·a_1_0·a_1_2 − c_2_44·c_2_5·a_1_0·a_1_1
       + c_2_3·c_2_54·a_1_0·a_1_2 + c_2_3·c_2_54·a_1_0·a_1_1
       + c_2_3·c_2_4·c_2_53·a_1_0·a_1_2 + c_2_3·c_2_4·c_2_53·a_1_0·a_1_1
       + c_2_3·c_2_42·c_2_52·a_1_0·a_1_2 + c_2_3·c_2_42·c_2_52·a_1_0·a_1_1
       + c_2_3·c_2_43·c_2_5·a_1_0·a_1_1 + c_2_3·c_2_44·a_1_1·a_1_2
       − c_2_3·c_2_44·a_1_0·a_1_2 − c_2_32·c_2_53·a_1_0·a_1_1
       − c_2_32·c_2_4·c_2_52·a_1_1·a_1_2 + c_2_32·c_2_42·c_2_5·a_1_1·a_1_2
       + c_2_32·c_2_42·c_2_5·a_1_0·a_1_1 − c_2_32·c_2_43·a_1_1·a_1_2
       + c_2_33·c_2_52·a_1_1·a_1_2 − c_2_33·c_2_52·a_1_0·a_1_2
       − c_2_33·c_2_52·a_1_0·a_1_1 − c_2_33·c_2_4·c_2_5·a_1_1·a_1_2
       + c_2_33·c_2_4·c_2_5·a_1_0·a_1_2 − c_2_33·c_2_4·c_2_5·a_1_0·a_1_1
       + c_2_33·c_2_42·a_1_1·a_1_2 − c_2_33·c_2_42·a_1_0·a_1_2
       + c_2_34·c_2_5·a_1_0·a_1_1 − c_2_34·c_2_4·a_1_0·a_1_2 − c_2_35·a_1_1·a_1_2, an element of degree 12
  13. c_12_0 → c_2_42·c_2_54 + c_2_43·c_2_53 + c_2_44·c_2_52 + c_2_3·c_2_4·c_2_54
       + c_2_3·c_2_42·c_2_53 + c_2_3·c_2_44·c_2_5 + c_2_32·c_2_54
       + c_2_32·c_2_4·c_2_53 + c_2_32·c_2_42·c_2_52 + c_2_32·c_2_44
       − c_2_33·c_2_43 + c_2_34·c_2_52 − c_2_34·c_2_4·c_2_5 − c_2_35·c_2_4 + c_2_36, an element of degree 12
  14. a_13_0 → c_2_4·c_2_55·a_1_1 − c_2_42·c_2_54·a_1_2 + c_2_42·c_2_54·a_1_1
       − c_2_43·c_2_53·a_1_2 − c_2_43·c_2_53·a_1_1 − c_2_43·c_2_53·a_1_0
       + c_2_44·c_2_52·a_1_2 − c_2_44·c_2_52·a_1_1 + c_2_45·c_2_5·a_1_2
       + c_2_45·c_2_5·a_1_0 − c_2_3·c_2_55·a_1_0 + c_2_3·c_2_4·c_2_54·a_1_0
       − c_2_3·c_2_42·c_2_53·a_1_1 + c_2_3·c_2_42·c_2_53·a_1_0
       − c_2_3·c_2_43·c_2_52·a_1_2 − c_2_3·c_2_43·c_2_52·a_1_1
       − c_2_3·c_2_43·c_2_52·a_1_0 + c_2_3·c_2_44·c_2_5·a_1_2 + c_2_3·c_2_44·c_2_5·a_1_1
       + c_2_3·c_2_45·a_1_2 + c_2_32·c_2_54·a_1_2 + c_2_32·c_2_54·a_1_1
       + c_2_32·c_2_4·c_2_53·a_1_2 − c_2_32·c_2_4·c_2_53·a_1_1
       + c_2_32·c_2_4·c_2_53·a_1_0 + c_2_32·c_2_43·c_2_5·a_1_2
       − c_2_32·c_2_43·c_2_5·a_1_1 − c_2_32·c_2_43·c_2_5·a_1_0 + c_2_32·c_2_44·a_1_2
       + c_2_33·c_2_53·a_1_1 + c_2_33·c_2_53·a_1_0 + c_2_33·c_2_4·c_2_52·a_1_2
       + c_2_33·c_2_4·c_2_52·a_1_1 − c_2_33·c_2_42·c_2_5·a_1_2
       − c_2_33·c_2_42·c_2_5·a_1_0 + c_2_33·c_2_43·a_1_2 − c_2_34·c_2_52·a_1_2
       − c_2_34·c_2_52·a_1_1 + c_2_34·c_2_4·c_2_5·a_1_2 − c_2_34·c_2_4·c_2_5·a_1_1
       − c_2_34·c_2_42·a_1_2 − c_2_35·c_2_5·a_1_1 + c_2_35·c_2_4·a_1_2, an element of degree 13
  15. a_17_1 → c_2_42·c_2_56·a_1_2 + c_2_42·c_2_56·a_1_1 − c_2_43·c_2_55·a_1_0
       + c_2_44·c_2_54·a_1_2 + c_2_44·c_2_54·a_1_1 + c_2_44·c_2_54·a_1_0
       + c_2_45·c_2_53·a_1_0 + c_2_46·c_2_52·a_1_2 + c_2_46·c_2_52·a_1_1
       − c_2_46·c_2_52·a_1_0 − c_2_3·c_2_43·c_2_54·a_1_2 − c_2_3·c_2_43·c_2_54·a_1_1
       − c_2_3·c_2_44·c_2_53·a_1_2 − c_2_3·c_2_44·c_2_53·a_1_1
       − c_2_3·c_2_44·c_2_53·a_1_0 − c_2_3·c_2_46·c_2_5·a_1_2 − c_2_3·c_2_46·c_2_5·a_1_1
       + c_2_3·c_2_46·c_2_5·a_1_0 − c_2_32·c_2_56·a_1_2 − c_2_32·c_2_56·a_1_1
       + c_2_32·c_2_43·c_2_53·a_1_2 + c_2_32·c_2_43·c_2_53·a_1_1
       + c_2_33·c_2_55·a_1_1 + c_2_33·c_2_4·c_2_54·a_1_2 + c_2_33·c_2_42·c_2_53·a_1_2
       − c_2_33·c_2_42·c_2_53·a_1_1 + c_2_33·c_2_43·c_2_52·a_1_1
       + c_2_33·c_2_44·c_2_5·a_1_2 − c_2_33·c_2_45·a_1_2 − c_2_34·c_2_54·a_1_2
       − c_2_34·c_2_54·a_1_1 − c_2_34·c_2_4·c_2_53·a_1_2 − c_2_34·c_2_43·c_2_5·a_1_2
       − c_2_34·c_2_43·c_2_5·a_1_1 − c_2_34·c_2_44·a_1_2 + c_2_35·c_2_53·a_1_1
       − c_2_35·c_2_43·a_1_2 − c_2_36·c_2_52·a_1_2 − c_2_36·c_2_52·a_1_1
       + c_2_36·c_2_4·c_2_5·a_1_2 + c_2_36·c_2_42·a_1_2 + c_2_37·c_2_5·a_1_1
       − c_2_37·c_2_4·a_1_2, an element of degree 17
  16. b_18_0 → c_2_42·c_2_57 + c_2_43·c_2_56 + c_2_44·c_2_55 + c_2_45·c_2_54
       + c_2_46·c_2_53 + c_2_47·c_2_52 + c_2_3·c_2_43·c_2_55 − c_2_3·c_2_44·c_2_54
       + c_2_3·c_2_46·c_2_52 − c_2_3·c_2_47·c_2_5 − c_2_32·c_2_57
       − c_2_32·c_2_4·c_2_56 + c_2_32·c_2_43·c_2_54 + c_2_32·c_2_46·c_2_5
       − c_2_33·c_2_4·c_2_55 + c_2_33·c_2_42·c_2_54 − c_2_33·c_2_43·c_2_53
       − c_2_33·c_2_44·c_2_52 − c_2_33·c_2_45·c_2_5 − c_2_34·c_2_55
       + c_2_34·c_2_4·c_2_54 − c_2_34·c_2_43·c_2_52 + c_2_34·c_2_44·c_2_5
       + c_2_35·c_2_4·c_2_53 − c_2_35·c_2_43·c_2_5 − c_2_36·c_2_53
       + c_2_36·c_2_42·c_2_5, an element of degree 18

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