Cohomology of group number 10 of order 81

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 81

General information on the group

  • The group has 2 minimal generators and exponent 9.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t3  −  t2  −  1)
    (t  −  1)2 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 81

Ring generators

The cohomology ring has 11 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. a_2_2, a nilpotent element of degree 2
  6. a_3_2, a nilpotent element of degree 3
  7. b_4_1, an element of degree 4
  8. a_5_1, a nilpotent element of degree 5
  9. b_6_1, an element of degree 6
  10. c_6_2, a Duflot regular element of degree 6
  11. a_7_3, a nilpotent element of degree 7

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 81

Ring relations

There are 5 "obvious" relations:
   a_1_02, a_1_12, a_3_22, a_5_12, a_7_32

Apart from that, there are 39 minimal relations of maximal degree 13:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_0
  3. a_2_1·a_1_0 − a_2_0·a_1_1
  4. a_2_2·a_1_1 + a_2_1·a_1_1 − a_2_0·a_1_1
  5. a_2_2·a_1_0 − a_2_1·a_1_1 + a_2_0·a_1_1
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_22 + a_2_12
  9.  − a_2_12 + a_2_0·a_2_2
  10. a_2_1·a_2_2 + a_2_12
  11. a_1_1·a_3_2 − a_2_12
  12. a_1_0·a_3_2
  13. a_2_2·a_3_2
  14. a_2_0·a_3_2
  15. a_2_1·a_3_2
  16. b_4_1·a_1_0
  17. a_2_0·b_4_1
  18. a_2_1·b_4_1
  19.  − a_2_2·b_4_1 + a_1_1·a_5_1
  20. a_1_0·a_5_1
  21. a_2_2·a_5_1
  22. a_2_0·a_5_1
  23. a_2_1·a_5_1
  24. b_6_1·a_1_1 + b_4_1·a_3_2
  25. b_6_1·a_1_0
  26. a_2_2·b_6_1 + a_3_2·a_5_1
  27. a_2_0·b_6_1
  28. a_2_1·b_6_1
  29. a_3_2·a_5_1 + a_1_1·a_7_3
  30. a_1_0·a_7_3
  31. b_6_1·a_3_2 − b_4_12·a_1_1
  32. a_2_2·a_7_3
  33. a_2_0·a_7_3
  34. a_2_1·a_7_3
  35. a_3_2·a_7_3 − b_4_1·a_1_1·a_5_1
  36.  − b_6_1·a_5_1 + b_4_1·a_7_3 − b_4_12·a_3_2
  37. b_6_12 + b_4_13 + a_5_1·a_7_3
  38. b_6_12 + b_4_13 + b_4_1·a_1_1·a_7_3
  39. b_6_1·a_7_3 + b_4_12·a_5_1 − b_4_13·a_1_1

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 81

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_6_2, a Duflot regular element of degree 6
    2.  − b_4_1, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 81

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  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. a_2_2 → 0, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. b_4_1 → 0, an element of degree 4
  8. a_5_1 → 0, an element of degree 5
  9. b_6_1 → 0, an element of degree 6
  10. c_6_2 →  − c_2_03, an element of degree 6
  11. a_7_3 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. 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. a_2_2 → 0, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. b_4_1 →  − c_2_22, an element of degree 4
  8. a_5_1 → c_2_22·a_1_1, an element of degree 5
  9. b_6_1 → c_2_23, an element of degree 6
  10. c_6_2 → c_2_1·c_2_22 − c_2_13, an element of degree 6
  11. a_7_3 →  − c_2_23·a_1_1, an element of degree 7

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 81


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