Cohomology of group number 21 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. a_3_7, a nilpotent element of degree 3
  10. b_3_6, 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_1 + a_1_0·a_3_5
  13. a_1_1·a_3_7
  14. a_2_0·b_2_2 + a_2_0·b_2_1 + a_1_0·a_3_7
  15. a_1_1·b_3_6
  16. a_1_0·b_3_6
  17. a_1_1·b_3_8 + a_2_0·b_2_3 + a_2_0·b_2_2
  18. a_1_0·b_3_8 + a_2_0·b_2_2 + a_2_0·b_2_1
  19. a_2_0·a_3_5
  20. b_2_2·a_3_5 + b_2_1·a_3_7 + b_2_1·a_3_5 + b_2_12·a_1_1
  21. b_2_3·a_3_5 + b_2_2·a_3_7 + b_2_2·a_3_5 + b_2_12·a_1_1
  22. a_2_0·a_3_7
  23. a_2_0·b_3_6
  24. b_2_2·b_3_6 + b_2_1·b_3_8 + b_2_1·b_3_6 + b_2_2·a_3_5 + b_2_1·a_3_5
  25. b_2_3·b_3_6 + b_2_2·b_3_8 + b_2_2·b_3_6 + b_2_3·a_3_5 + b_2_2·a_3_5
  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 + b_2_1·a_1_0·a_3_5
  30. a_3_72 + b_2_1·a_1_0·a_3_7
  31. a_3_5·a_3_7 + b_2_1·a_1_0·a_3_5
  32. b_3_62 + b_2_1·b_2_32 + b_2_12·b_2_3
  33. a_3_7·b_3_6 + a_3_5·b_3_8 + b_2_1·a_1_0·a_3_7
  34. 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_7·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_6 + b_2_1·a_4_4 + b_2_1·a_1_0·a_3_7 + b_2_1·a_1_0·a_3_5
  37. a_3_7·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_7·b_3_6 + a_3_5·b_3_6 + b_2_2·a_4_4 + b_2_1·a_1_0·a_3_7 + b_2_1·a_1_0·a_3_5
  39. a_2_0·a_4_4
  40. a_4_4·a_3_7
  41. a_4_4·a_3_5
  42. a_4_4·b_3_6 + b_2_2·b_2_3·a_3_7 + b_2_1·b_2_3·a_3_7
  43. a_4_4·b_3_8 + b_2_32·a_3_7 + b_2_1·b_2_3·a_3_7
  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_02, an element of degree 2
  8. a_3_5 → 0, an element of degree 3
  9. a_3_7 → 0, an element of degree 3
  10. b_3_6 → 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_14 + 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_22, 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_32, an element of degree 2
  7. c_2_4 → c_1_32 + c_1_02, an element of degree 2
  8. a_3_5 → 0, an element of degree 3
  9. a_3_7 → 0, an element of degree 3
  10. b_3_6 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  11. b_3_8 → c_1_33 + c_1_22·c_1_3, an element of degree 3
  12. a_4_4 → 0, an element of degree 4
  13. c_4_13 → c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14
       + 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

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


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