Cohomology of group number 45 of order 243

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General information on the group

  • The group has 3 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
    t8  −  t7  +  2·t6  −  2·t5  +  t4  −  t3  −  1
    (t  −  1)3 · (t2  −  t  +  1)2 · (t2  +  t  +  1)2
  • 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 14 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_1_2, a nilpotent element of degree 1
  4. b_2_3, an element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. a_3_3, a nilpotent element of degree 3
  7. a_3_4, a nilpotent element of degree 3
  8. a_5_3, a nilpotent element of degree 5
  9. a_5_4, a nilpotent element of degree 5
  10. a_5_5, a nilpotent element of degree 5
  11. a_5_6, a nilpotent element of degree 5
  12. c_6_8, a Duflot regular element of degree 6
  13. c_6_9, a Duflot regular element of degree 6
  14. a_7_12, a nilpotent element of degree 7

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Ring relations

There are 11 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_22, a_3_32, a_3_42, a_5_32, a_5_42, a_5_52, a_5_62, a_7_122

Apart from that, there are 44 minimal relations of maximal degree 12:

  1. b_2_3·a_1_1
  2. b_2_3·a_1_2
  3. a_1_1·a_3_2
  4. a_1_2·a_3_2 + a_1_1·a_3_3
  5. a_1_2·a_3_3 − a_1_2·a_3_2 + a_1_1·a_3_4
  6. a_1_2·a_3_4 + a_1_2·a_3_3 + a_1_2·a_3_2
  7. b_2_3·a_3_2
  8. b_2_3·a_3_3
  9. b_2_3·a_3_4
  10. a_1_1·a_5_3
  11.  − a_3_2·a_3_3 + a_1_2·a_5_3
  12. a_3_2·a_3_3 + a_1_1·a_5_4
  13. a_3_2·a_3_4 − a_3_2·a_3_3 + a_1_2·a_5_4
  14.  − a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_1·a_5_5
  15.  − a_3_3·a_3_4 − a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_2·a_5_5
  16. a_3_3·a_3_4 − a_3_2·a_3_3 + a_1_1·a_5_6
  17. a_1_2·a_5_6
  18. b_2_3·a_5_4 + b_2_3·a_5_3
  19. b_2_3·a_5_5 + b_2_3·a_5_3
  20. a_3_2·a_5_3
  21.  − a_3_4·a_5_3 + a_3_3·a_5_4
  22. a_3_3·a_5_3 + a_3_2·a_5_4
  23.  − a_3_4·a_5_4 − a_3_4·a_5_3 + a_3_3·a_5_5 − a_3_3·a_5_3
  24.  − a_3_4·a_5_3 + a_3_3·a_5_3 + a_3_2·a_5_5
  25. a_3_4·a_5_5 − a_3_4·a_5_4 − a_3_4·a_5_3 + a_3_3·a_5_6 + a_3_3·a_5_3
  26.  − a_3_4·a_5_4 + a_3_4·a_5_3 + a_3_2·a_5_6
  27. a_3_4·a_5_6 − a_3_4·a_5_5 − a_3_4·a_5_4 − a_3_3·a_5_3
  28.  − a_3_4·a_5_5 + a_3_4·a_5_4 + a_3_4·a_5_3 − a_3_3·a_5_3 + c_6_8·a_1_1·a_1_2
  29. a_3_4·a_5_5 − a_3_4·a_5_4 − a_3_4·a_5_3 − a_3_3·a_5_3 + c_6_9·a_1_1·a_1_2
  30. a_3_4·a_5_5 − a_3_4·a_5_4 + a_3_3·a_5_3 + a_1_1·a_7_12 − c_6_8·a_1_0·a_1_1
  31.  − a_3_4·a_5_5 − a_3_4·a_5_4 + a_1_2·a_7_12 − c_6_8·a_1_0·a_1_2
  32. b_2_3·a_7_12 + b_2_32·a_5_6 + b_2_3·c_6_8·a_1_0
  33. a_5_4·a_5_5 + a_5_3·a_5_5 − a_5_3·a_5_4
  34. a_5_4·a_5_6 + a_5_3·a_5_6 + c_6_8·a_1_1·a_3_3
  35.  − a_5_5·a_5_6 + a_5_4·a_5_6 + c_6_8·a_1_1·a_3_4
  36.  − a_5_4·a_5_6 − a_5_3·a_5_6 + a_5_3·a_5_4 + c_6_9·a_1_1·a_3_3
  37. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_3·a_5_5 + a_5_3·a_5_4 + c_6_9·a_1_1·a_3_4
  38. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_3·a_5_5 − a_5_3·a_5_4 + a_3_3·a_7_12 − c_6_8·a_1_0·a_3_3
  39.  − a_5_4·a_5_6 − a_5_3·a_5_6 − a_5_3·a_5_5 + a_5_3·a_5_4 + a_3_2·a_7_12 − c_6_8·a_1_0·a_3_2
  40. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_3·a_5_5 − a_5_3·a_5_4 + a_3_4·a_7_12 − c_6_8·a_1_0·a_3_4
  41. a_5_3·a_7_12 + b_2_3·a_5_3·a_5_6 + c_6_9·a_1_1·a_5_5 − c_6_9·a_1_1·a_5_4
       + c_6_8·a_1_1·a_5_5 − c_6_8·a_1_0·a_5_3
  42. a_5_5·a_7_12 − b_2_3·a_5_3·a_5_6 − c_6_9·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_5
       − c_6_8·a_1_0·a_5_5
  43. a_5_4·a_7_12 − b_2_3·a_5_3·a_5_6 − c_6_9·a_1_1·a_5_6 + c_6_9·a_1_1·a_5_4
       − c_6_8·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_5 − c_6_8·a_1_0·a_5_4
  44. a_5_6·a_7_12 − c_6_9·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_5
       + c_6_8·a_1_1·a_5_4 − c_6_8·a_1_0·a_5_6

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_8, a Duflot regular element of degree 6
    2. c_6_9, a Duflot regular element of degree 6
    3. b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_5_3 → 0, an element of degree 5
  9. a_5_4 → 0, an element of degree 5
  10. a_5_5 → 0, an element of degree 5
  11. a_5_6 → 0, an element of degree 5
  12. c_6_8 →  − c_2_23, an element of degree 6
  13. c_6_9 → c_2_23 + c_2_13, an element of degree 6
  14. a_7_12 → 0, an element of degree 7

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

  1. a_1_0 → a_1_2, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. b_2_3 → c_2_5, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_5_3 → c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
  9. a_5_4 →  − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  10. a_5_5 →  − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  11. a_5_6 →  − c_2_52·a_1_0 + c_2_3·c_2_5·a_1_2, an element of degree 5
  12. c_6_8 →  − c_2_52·a_1_1·a_1_2 + c_2_4·c_2_52 − c_2_43, an element of degree 6
  13. c_6_9 →  − c_2_52·a_1_0·a_1_2 − c_2_4·c_2_52 + c_2_43 − c_2_3·c_2_52 + c_2_33, an element of degree 6
  14. a_7_12 → c_2_53·a_1_0 − c_2_4·c_2_52·a_1_2 + c_2_43·a_1_2 − c_2_3·c_2_52·a_1_2, an element of degree 7

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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