Cohomology of group number 142 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 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) · (t5  −  t4  +  t2  −  t  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-4,-3,-3. 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 8:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. a_2_2, a nilpotent element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. b_3_3, an element of degree 3
  7. b_3_4, an element of degree 3
  8. b_4_5, an element of degree 4
  9. a_5_2, a nilpotent element of degree 5
  10. b_5_6, an element of degree 5
  11. b_6_8, an element of degree 6
  12. b_7_10, an element of degree 7
  13. c_8_13, a Duflot regular element of degree 8

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Ring relations

There are 51 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_1·a_1_0
  4. a_2_1·b_1_1 + a_2_2·a_1_0
  5. a_2_2·b_1_1 + a_2_1·b_1_1
  6. a_2_22 + a_2_1·a_2_2
  7. a_1_0·a_3_2 + a_2_12
  8. b_1_1·a_3_2 + a_2_22
  9. a_1_0·b_3_3 + a_2_12
  10. a_1_0·b_3_4 + a_2_22 + a_2_12
  11. a_2_1·a_3_2
  12. a_2_2·b_3_3 + a_2_2·a_3_2
  13. a_2_1·b_3_3
  14. a_2_2·b_3_4
  15. a_2_1·b_3_4 + a_2_2·a_3_2
  16. b_4_5·a_1_0
  17. a_3_2·b_3_3 + a_3_22
  18. a_3_2·b_3_4 + a_2_2·b_4_5 + a_3_22
  19. a_2_1·b_4_5 + a_3_22
  20. b_3_32 + b_4_5·b_1_12 + a_3_22
  21. a_1_0·a_5_2
  22. b_1_1·a_5_2
  23. a_1_0·b_5_6
  24. b_3_42 + b_1_1·b_5_6 + b_1_13·b_3_3 + a_3_2·b_3_4
  25. a_2_1·a_5_2
  26. a_2_2·b_5_6
  27. a_2_1·b_5_6 + a_2_2·a_5_2
  28. b_6_8·a_1_0
  29. b_1_12·b_5_6 + b_1_14·b_3_4 + b_6_8·b_1_1 + b_4_5·b_3_3 + b_4_5·a_3_2 + a_2_2·a_5_2
  30. b_3_3·a_5_2 + a_3_2·a_5_2
  31. b_3_4·a_5_2 + a_3_2·b_5_6
  32. b_3_4·a_5_2 + a_2_2·b_6_8 + a_3_2·a_5_2
  33. a_2_1·b_6_8 + a_3_2·a_5_2
  34. a_1_0·b_7_10
  35. b_3_3·b_5_6 + b_1_1·b_7_10 + b_1_15·b_3_4 + b_6_8·b_1_12 + b_4_5·b_1_1·b_3_4
       + b_4_5·b_1_1·b_3_3 + b_4_5·b_1_14 + b_3_4·a_5_2
  36. b_6_8·a_3_2 + b_4_5·a_5_2 + a_2_2·b_4_5·a_3_2
  37. a_2_2·b_7_10
  38. a_2_1·b_7_10
  39. b_1_12·b_7_10 + b_1_13·b_3_3·b_3_4 + b_1_16·b_3_4 + b_6_8·b_3_3 + b_6_8·b_1_13
       + b_4_5·b_1_12·b_3_4 + b_4_5·b_1_12·b_3_3 + b_4_5·b_1_15 + b_4_52·b_1_1
       + b_4_5·a_5_2
  40. a_5_22 + b_4_5·a_3_22
  41. a_5_2·b_5_6 + a_2_2·b_4_52 + b_4_5·a_3_22
  42. a_3_2·b_7_10
  43. b_3_3·b_7_10 + b_1_14·b_3_3·b_3_4 + b_6_8·b_1_1·b_3_3 + b_4_5·b_3_3·b_3_4
       + b_4_5·b_1_1·b_5_6 + b_4_5·b_1_13·b_3_3 + b_4_52·b_1_12 + a_2_2·b_4_52
       + b_4_5·a_3_22
  44. b_5_62 + b_1_17·b_3_4 + b_1_17·b_3_3 + b_6_8·b_1_1·b_3_3 + b_4_5·b_1_1·b_5_6
       + b_4_5·b_1_13·b_3_4 + a_2_2·b_4_52 + b_4_5·a_3_22 + c_8_13·b_1_12
  45. b_6_8·a_5_2 + b_4_52·a_3_2 + a_2_2·b_4_5·a_5_2
  46. b_6_8·b_5_6 + b_6_8·b_1_12·b_3_4 + b_6_8·b_1_12·b_3_3 + b_6_8·b_1_15 + b_4_5·b_7_10
       + b_4_5·b_1_1·b_3_3·b_3_4 + b_4_5·b_1_14·b_3_4 + b_4_5·b_1_14·b_3_3 + b_4_52·b_3_4
       + b_4_52·b_1_13 + c_8_13·b_1_13 + a_2_2·c_8_13·a_1_0
  47. a_5_2·b_7_10
  48. b_6_8·b_1_13·b_3_3 + b_6_8·b_1_16 + b_6_82 + b_4_5·b_1_15·b_3_3
       + b_4_5·b_6_8·b_1_12 + b_4_52·b_1_1·b_3_3 + b_4_53 + b_4_5·a_3_2·a_5_2
       + c_8_13·b_1_14 + a_2_12·c_8_13
  49. b_5_6·b_7_10 + b_1_16·b_3_3·b_3_4 + b_1_19·b_3_4 + b_1_19·b_3_3 + b_6_8·b_1_16
       + b_6_82 + b_4_5·b_3_4·b_5_6 + b_4_5·b_1_1·b_7_10 + b_4_5·b_1_12·b_3_3·b_3_4
       + b_4_5·b_1_15·b_3_3 + b_4_5·b_1_18 + b_4_5·b_6_8·b_1_12 + b_4_52·b_1_1·b_3_4
       + b_4_52·b_1_14 + b_4_53 + a_2_2·b_4_5·b_6_8 + c_8_13·b_1_1·b_3_3
       + a_2_1·a_2_2·c_8_13 + a_2_12·c_8_13
  50. b_6_8·b_7_10 + b_6_8·b_1_1·b_3_3·b_3_4 + b_6_8·b_1_14·b_3_4 + b_6_8·b_1_17
       + b_4_5·b_1_16·b_3_3 + b_4_5·b_6_8·b_3_4 + b_4_5·b_6_8·b_1_13 + b_4_52·b_5_6
       + b_4_52·b_1_12·b_3_4 + b_4_52·b_1_12·b_3_3 + b_4_52·b_1_15
       + a_2_2·b_4_52·a_3_2 + c_8_13·b_1_12·b_3_3 + c_8_13·b_1_15
  51. b_7_102 + b_1_111·b_3_4 + b_1_111·b_3_3 + b_6_8·b_1_18 + b_6_82·b_1_12
       + b_4_5·b_1_17·b_3_4 + b_4_5·b_6_8·b_1_1·b_3_3 + b_4_52·b_1_13·b_3_4
       + b_4_52·b_1_13·b_3_3 + b_4_52·b_1_16 + b_4_53·b_1_12 + b_4_5·c_8_13·b_1_12

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 14.
  • 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_5, an element of degree 4
    3. b_3_3, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

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. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → 0, an element of degree 3
  7. b_3_4 → 0, an element of degree 3
  8. b_4_5 → 0, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. b_5_6 → 0, an element of degree 5
  11. b_6_8 → 0, an element of degree 6
  12. b_7_10 → 0, an element of degree 7
  13. c_8_13 → c_1_08, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_3_4 → c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  8. b_4_5 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. b_5_6 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. b_6_8 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22
       + c_1_15·c_1_2 + c_1_0·c_1_15 + c_1_04·c_1_12, an element of degree 6
  12. b_7_10 → c_1_15·c_1_22 + c_1_16·c_1_2 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_02·c_1_15
       + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2 + c_1_04·c_1_13, an element of degree 7
  13. c_8_13 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + c_1_0·c_1_17
       + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14
       + c_1_08, an element of degree 8

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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