Cohomology of group number 897 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 16.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t4  −  t2  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-5,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 11 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, an element of degree 1
  4. a_3_3, a nilpotent element of degree 3
  5. b_4_4, an element of degree 4
  6. a_5_5, a nilpotent element of degree 5
  7. a_5_6, a nilpotent element of degree 5
  8. b_6_8, an element of degree 6
  9. a_7_10, a nilpotent element of degree 7
  10. c_8_13, a Duflot regular element of degree 8
  11. a_9_11, a nilpotent element of degree 9

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Ring relations

There are 34 minimal relations of maximal degree 18:

  1. a_1_02
  2. a_1_0·b_1_1 + a_1_22
  3. a_1_22·b_1_1
  4. a_1_0·a_3_3
  5. b_1_12·a_3_3
  6. b_4_4·a_1_0
  7. a_3_32
  8. a_1_0·a_5_5
  9. a_1_0·a_5_6
  10. b_1_12·a_5_6 + b_1_12·a_5_5
  11. b_6_8·a_1_0
  12. a_3_3·a_5_5
  13. a_3_3·a_5_6
  14. a_1_0·a_7_10
  15. b_6_8·a_3_3 + b_4_4·a_5_6 + b_4_4·a_5_5 + b_4_4·a_1_2·b_1_1·a_3_3
  16. b_1_12·a_7_10 + b_1_14·a_5_5 + b_6_8·a_1_2·b_1_12 + b_4_4·a_5_5
       + b_4_4·a_1_2·b_1_14 + a_1_2·b_1_13·a_5_5
  17. a_5_52
  18. a_5_5·a_5_6
  19. a_5_62
  20. a_3_3·a_7_10
  21. a_1_0·a_9_11
  22. b_6_8·a_5_6 + b_6_8·a_5_5 + b_4_42·a_3_3 + b_4_4·a_1_2·b_1_1·a_5_6
       + b_4_4·a_1_2·b_1_1·a_5_5
  23. b_1_12·a_9_11 + b_1_16·a_5_5 + b_6_8·a_5_5 + b_4_4·a_1_2·b_1_16
       + b_4_42·a_1_2·b_1_12
  24. a_5_5·a_7_10 + b_6_8·a_1_2·a_5_5 + b_4_4·a_1_2·b_1_12·a_5_5
  25. a_5_6·a_7_10 + b_6_8·a_1_2·a_5_5 + b_4_4·a_1_2·b_1_12·a_5_5
  26. b_6_8·b_1_16 + b_6_82 + b_4_4·b_1_18 + b_4_4·b_6_8·b_1_12 + b_4_42·b_1_14
       + b_4_43 + b_4_4·b_1_13·a_5_5 + b_4_42·b_1_1·a_3_3 + a_1_2·b_1_16·a_5_5
       + b_4_4·a_1_2·b_1_12·a_5_5 + c_8_13·b_1_14
  27. a_3_3·a_9_11
  28. b_6_8·a_7_10 + b_6_8·b_1_12·a_5_5 + b_6_82·a_1_2 + b_4_4·a_9_11 + b_4_4·b_1_14·a_5_5
       + b_4_4·b_6_8·a_1_2·b_1_12 + b_4_42·a_1_2·b_1_14 + b_4_43·a_1_2
       + b_6_8·a_1_2·b_1_1·a_5_5 + b_4_42·a_1_2·b_1_1·a_3_3
  29. a_7_102
  30. a_5_5·a_9_11 + b_4_4·a_1_2·b_1_14·a_5_5 + b_4_42·a_1_2·a_5_5
  31. a_5_6·a_9_11 + b_4_4·a_1_2·b_1_14·a_5_5 + b_4_42·a_1_2·a_5_5
  32. b_6_8·a_9_11 + b_4_4·b_1_16·a_5_5 + b_4_4·b_6_8·a_5_5 + b_4_4·b_6_8·a_1_2·b_1_14
       + b_4_42·a_7_10 + b_4_43·a_1_2·b_1_12 + b_4_42·a_1_2·b_1_1·a_5_5
       + c_8_13·b_1_12·a_5_5
  33. a_7_10·a_9_11 + b_4_4·a_1_2·b_1_16·a_5_5 + b_4_42·a_1_2·b_1_12·a_5_5
       + c_8_13·a_1_2·b_1_12·a_5_5
  34. a_9_112

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 18.
  • 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_8_13, a Duflot regular element of degree 8
    2. b_1_14 + b_4_4, an element of degree 4
    3. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 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 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. b_4_4 → 0, an element of degree 4
  6. a_5_5 → 0, an element of degree 5
  7. a_5_6 → 0, an element of degree 5
  8. b_6_8 → 0, an element of degree 6
  9. a_7_10 → 0, an element of degree 7
  10. c_8_13 → c_1_08, an element of degree 8
  11. a_9_11 → 0, an element of degree 9

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → c_1_1, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. b_4_4 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  6. a_5_5 → 0, an element of degree 5
  7. a_5_6 → 0, an element of degree 5
  8. b_6_8 → c_1_26 + c_1_14·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  9. a_7_10 → 0, an element of degree 7
  10. c_8_13 → c_1_28 + c_1_12·c_1_26 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8
  11. a_9_11 → 0, an element of degree 9

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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