Cohomology of group number 14 of order 128

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General information on the group

  • The group has 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1) · (t3  −  2·t2  +  t  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 4:

  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. b_2_1, an element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. c_2_4, a Duflot regular element of degree 2
  8. a_3_5, a nilpotent element of degree 3
  9. b_3_6, an element of degree 3
  10. b_3_7, an element of degree 3
  11. b_3_8, an element of degree 3
  12. a_4_4, a nilpotent element of degree 4
  13. c_4_13, a Duflot regular element of degree 4

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Ring relations

There are 44 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_1
  5. a_2_0·a_1_0
  6. b_2_2·a_1_1 + b_2_1·a_1_1
  7. b_2_2·a_1_0 + b_2_1·a_1_1
  8. b_2_3·a_1_0 + b_2_1·a_1_1
  9. a_2_02
  10. b_2_22 + b_2_1·b_2_3
  11. a_2_0·b_2_2 + a_1_1·a_3_5
  12. a_2_0·b_2_2 + a_1_0·a_3_5
  13. a_1_1·b_3_6
  14. a_1_0·b_3_6
  15. a_1_1·b_3_7
  16. a_1_0·b_3_7 + a_2_0·b_2_2 + a_2_0·b_2_1
  17. a_1_1·b_3_8 + a_2_0·b_2_3
  18. a_1_0·b_3_8 + a_2_0·b_2_2
  19. a_2_0·a_3_5
  20. a_2_0·b_3_6
  21. b_2_3·b_3_7 + b_2_2·b_3_6 + b_2_3·a_3_5 + b_2_2·a_3_5 + b_2_12·a_1_1
  22. b_2_2·b_3_7 + b_2_1·b_3_6 + b_2_2·a_3_5 + b_2_1·a_3_5 + b_2_12·a_1_1
  23. b_2_12·a_1_1 + b_2_12·a_1_0 + a_2_0·b_3_7
  24. b_2_2·b_3_6 + b_2_1·b_3_8 + b_2_1·b_3_6 + b_2_1·a_3_5 + b_2_12·a_1_1
  25. b_2_3·b_3_6 + b_2_2·b_3_8 + b_2_2·b_3_6 + b_2_2·a_3_5 + b_2_12·a_1_1
  26. b_2_32·a_1_1 + b_2_12·a_1_1 + a_2_0·b_3_8
  27. a_4_4·a_1_1
  28. a_4_4·a_1_0
  29. a_3_52
  30. b_3_62 + b_2_1·b_2_32 + b_2_12·b_2_3
  31. b_3_6·b_3_7 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_2 + a_3_5·b_3_7 + a_3_5·b_3_6
       + b_2_1·a_1_0·a_3_5
  32. b_3_72 + b_2_12·b_2_3 + b_2_13
  33. b_3_6·b_3_8 + b_2_2·b_2_32 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3
       + a_3_5·b_3_6
  34. b_3_7·b_3_8 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3 + b_2_12·b_2_2
       + a_3_5·b_3_8 + a_3_5·b_3_6
  35. b_3_82 + b_2_33 + b_2_12·b_2_3 + a_2_0·b_2_32 + b_2_1·a_1_0·a_3_5
  36. a_3_5·b_3_7 + b_2_1·a_4_4 + a_2_0·b_2_12
  37. a_3_5·b_3_8 + a_3_5·b_3_6 + b_2_3·a_4_4 + b_2_1·a_1_0·a_3_5
  38. a_3_5·b_3_6 + b_2_2·a_4_4
  39. a_2_0·a_4_4
  40. a_4_4·a_3_5
  41. a_4_4·b_3_6 + b_2_2·b_2_3·a_3_5 + b_2_1·b_2_2·a_3_5
  42. a_4_4·b_3_7 + b_2_1·b_2_3·a_3_5 + b_2_12·a_3_5 + a_2_0·b_2_1·b_3_7
  43. a_4_4·b_3_8 + b_2_32·a_3_5 + b_2_2·b_2_3·a_3_5 + b_2_1·b_2_3·a_3_5 + b_2_1·b_2_2·a_3_5
  44. a_4_42

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_2_4, a Duflot regular element of degree 2
    2. c_4_13, a Duflot regular element of degree 4
    3. b_2_3 + b_2_2 + b_2_1, an element of degree 2
    4. b_3_6, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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_2_0 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. c_2_4 → c_1_12, an element of degree 2
  8. a_3_5 → 0, an element of degree 3
  9. b_3_6 → 0, an element of degree 3
  10. b_3_7 → 0, an element of degree 3
  11. b_3_8 → 0, an element of degree 3
  12. a_4_4 → 0, an element of degree 4
  13. c_4_13 → c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  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. b_2_1 → c_1_32, an element of degree 2
  5. b_2_2 → c_1_2·c_1_3, an element of degree 2
  6. b_2_3 → c_1_22, an element of degree 2
  7. c_2_4 → c_1_32 + c_1_12, an element of degree 2
  8. a_3_5 → 0, an element of degree 3
  9. b_3_6 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  10. b_3_7 → c_1_33 + c_1_2·c_1_32, an element of degree 3
  11. b_3_8 → c_1_2·c_1_32 + c_1_23, an element of degree 3
  12. a_4_4 → 0, an element of degree 4
  13. c_4_13 → c_1_23·c_1_3 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4

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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