Mod-3-Cohomology of group number 39 of order 324

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General information on the group

  • The group order factors as 22 · 34.
  • 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*(Syl3(A9); 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  −  3·t  +  6·t2  −  7·t3  +  7·t4  −  6·t5  +  6·t6  −  6·t7  +  6·t8  −  5·t9  +  6·t10  −  6·t11  +  5·t12  −  3·t13  +  t14)
    ( − 1  +  t)3 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)3 · (1  −  t2  +  t4)
  • The a-invariants are -∞,-∞,-5,-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 14 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. b_4_2, an element of degree 4
  5. b_4_1, an element of degree 4
  6. b_4_0, an element of degree 4
  7. a_6_0, a nilpotent element of degree 6
  8. a_7_6, a nilpotent element of degree 7
  9. a_7_1, a nilpotent element of degree 7
  10. b_8_4, an element of degree 8
  11. a_10_6, a nilpotent element of degree 10
  12. a_11_13, a nilpotent element of degree 11
  13. a_11_9, a nilpotent element of degree 11
  14. c_12_7, a Duflot element of degree 12

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Ring relations

There are 7 "obvious" relations:
   a_3_02, a_3_12, a_3_22, a_7_12, a_7_62, a_11_92, a_11_132

Apart from that, there are 56 minimal relations of maximal degree 22:

  1. a_3_0·a_3_2
  2. a_3_1·a_3_2
  3. b_4_0·a_3_2
  4. b_4_1·a_3_2
  5. b_4_2·a_3_0
  6. b_4_2·a_3_1
  7. b_4_0·b_4_2
  8. b_4_1·b_4_2
  9. a_6_0·a_3_1
  10. a_6_0·a_3_2
  11. a_3_1·a_7_6
  12. a_3_2·a_7_1
  13. a_3_2·a_7_6
  14. b_4_1·a_6_0 − a_3_1·a_7_1
  15. b_4_2·a_6_0
  16. b_4_2·a_7_1
  17. b_4_2·a_7_6
  18. b_8_4·a_3_1 − b_4_1·a_7_6
  19. b_8_4·a_3_2
  20. a_6_02
  21. b_4_2·b_8_4
  22. a_6_0·a_7_1
  23. a_6_0·a_7_6
  24. a_10_6·a_3_0
  25. a_10_6·a_3_1
  26. a_10_6·a_3_2
  27. a_3_0·a_11_13
  28. a_3_1·a_11_13
  29. a_3_2·a_11_9
  30. a_7_1·a_7_6 + a_3_1·a_11_9
  31. b_4_0·a_10_6
  32. b_4_1·a_10_6
  33. b_4_2·a_10_6 − a_3_2·a_11_13
  34. a_6_0·b_8_4 − a_3_1·a_11_9
  35. b_4_0·a_11_13
  36. b_4_1·a_11_13
  37. b_4_2·a_11_9
  38. b_8_4·a_7_1 − b_4_1·a_11_9
  39. b_8_4·a_7_6 − c_12_7·a_3_1
  40. a_6_0·a_10_6
  41. b_8_42 − b_4_1·c_12_7
  42. a_6_0·a_11_9
  43. a_6_0·a_11_13
  44. a_10_6·a_7_1
  45. a_10_6·a_7_6
  46. a_7_1·a_11_9
  47. a_7_1·a_11_13
  48. a_7_6·a_11_9 − a_6_0·c_12_7
  49. a_7_6·a_11_13
  50. b_8_4·a_10_6
  51. b_8_4·a_11_9 − c_12_7·a_7_1
  52. b_8_4·a_11_13
  53. a_10_62
  54. a_10_6·a_11_9
  55. a_10_6·a_11_13
  56. a_11_9·a_11_13

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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_7, an element of degree 12
    2.  − b_4_23 + b_4_1·b_8_4 − b_4_13 − b_4_02·b_4_1 + b_4_03, an element of degree 12
    3. b_4_1, an element of degree 4
  • A Duflot regular sequence is given by c_12_7.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 19, 25].
  • We found that there exists some filter regular HSOP over a finite extension field, formed by c_12_7, together with 2 elements of degree 4.
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. c_12_7, an element of degree 12
    2. b_4_2 + b_4_0, an element of degree 4
    3. b_4_1, an element of degree 4

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Restriction maps

Expressing the generators as elements of H*(Syl3(A9); GF(3))

  1. a_3_2 → b_2_0·a_1_0
  2. a_3_1 → b_2_2·a_1_1
  3. a_3_0 → a_3_4
  4. b_4_2 → b_2_02
  5. b_4_1 → b_2_22
  6. b_4_0 → b_4_6
  7. a_6_0 → a_1_1·a_5_8
  8. a_7_6 → c_6_11·a_1_1
  9. a_7_1 → b_2_2·a_5_8
  10. b_8_4 → b_2_2·c_6_11
  11. a_10_6 → c_6_11·a_1_0·a_3_3
  12. a_11_13 → b_2_0·c_6_11·a_3_3
  13. a_11_9 → c_6_11·a_5_8
  14. c_12_7 → c_6_112

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. b_4_2 → 0, an element of degree 4
  5. b_4_1 → 0, an element of degree 4
  6. b_4_0 → 0, an element of degree 4
  7. a_6_0 → 0, an element of degree 6
  8. a_7_6 → 0, an element of degree 7
  9. a_7_1 → 0, an element of degree 7
  10. b_8_4 → 0, an element of degree 8
  11. a_10_6 → 0, an element of degree 10
  12. a_11_13 → 0, an element of degree 11
  13. a_11_9 → 0, an element of degree 11
  14. c_12_7 → 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 → 0, an element of degree 3
  3. a_3_0 → 0, an element of degree 3
  4. b_4_2 → c_2_22, an element of degree 4
  5. b_4_1 → 0, an element of degree 4
  6. b_4_0 → 0, an element of degree 4
  7. a_6_0 → 0, an element of degree 6
  8. a_7_6 → 0, an element of degree 7
  9. a_7_1 → 0, an element of degree 7
  10. b_8_4 → 0, an element of degree 8
  11. a_10_6 →  − 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
  12. a_11_13 → 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
  13. a_11_9 → 0, an element of degree 11
  14. c_12_7 → 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 3 in a Sylow subgroup

  1. a_3_2 → 0, an element of degree 3
  2. a_3_1 → c_2_5·a_1_2, an element of degree 3
  3. a_3_0 →  − c_2_5·a_1_1 + c_2_5·a_1_0 − c_2_4·a_1_2 − c_2_4·a_1_1 + c_2_3·a_1_2, an element of degree 3
  4. b_4_2 → 0, an element of degree 4
  5. b_4_1 → c_2_52, an element of degree 4
  6. b_4_0 → c_2_4·c_2_5 − c_2_42 − c_2_3·c_2_5, an element of degree 4
  7. a_6_0 →  − c_2_4·c_2_5·a_1_0·a_1_2 + c_2_42·a_1_0·a_1_2 − c_2_3·c_2_5·a_1_1·a_1_2
       + c_2_3·c_2_5·a_1_0·a_1_2 − c_2_3·c_2_4·a_1_1·a_1_2, an element of degree 6
  8. a_7_6 → c_2_3·c_2_4·c_2_5·a_1_2 − c_2_3·c_2_42·a_1_2 + c_2_32·c_2_5·a_1_2 + c_2_33·a_1_2, an element of degree 7
  9. 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_52·a_1_0
       + 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_32·c_2_5·a_1_2, an element of degree 7
  10. b_8_4 → c_2_3·c_2_4·c_2_52 − c_2_3·c_2_42·c_2_5 + c_2_32·c_2_52 + c_2_33·c_2_5, an element of degree 8
  11. a_10_6 → 0, an element of degree 10
  12. a_11_13 → 0, an element of degree 11
  13. a_11_9 → c_2_3·c_2_42·c_2_52·a_1_0 + c_2_3·c_2_43·c_2_5·a_1_0 + c_2_3·c_2_44·a_1_0
       + 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_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_5·a_1_0 + c_2_34·c_2_4·a_1_2 + c_2_34·c_2_4·a_1_1
       + c_2_35·a_1_2, an element of degree 11
  14. c_12_7 → c_2_32·c_2_42·c_2_52 + c_2_32·c_2_43·c_2_5 + c_2_32·c_2_44
       − c_2_33·c_2_4·c_2_52 + c_2_33·c_2_42·c_2_5 + c_2_34·c_2_52
       − c_2_34·c_2_4·c_2_5 + c_2_34·c_2_42 − c_2_35·c_2_5 + c_2_36, 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