Cohomology of group number 22 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 2.
  • 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
     − 1
    (t  +  1) · (t  −  1)3
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 10 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. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. c_2_3, a Duflot regular element of degree 2
  7. a_3_4, a nilpotent element of degree 3
  8. b_3_5, an element of degree 3
  9. a_4_4, a nilpotent element of degree 4
  10. c_4_8, a Duflot regular element of degree 4

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Ring relations

There are 27 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_0
  5. a_2_1·a_1_1
  6. a_2_1·a_1_0 + a_2_0·a_1_1
  7. b_2_2·a_1_0
  8. a_2_02
  9. a_2_0·a_2_1
  10. a_2_12
  11. a_2_0·b_2_2 + a_1_1·a_3_4
  12. a_1_0·a_3_4
  13. a_1_1·b_3_5 + a_2_1·b_2_2
  14. a_1_0·b_3_5
  15. a_2_0·a_3_4 + a_2_0·c_2_3·a_1_1
  16. a_2_0·b_3_5 + a_2_1·a_3_4
  17. b_2_22·a_1_1 + a_2_1·b_3_5
  18. a_4_4·a_1_1 + a_2_1·a_3_4
  19. a_4_4·a_1_0 + a_2_0·c_2_3·a_1_1
  20. a_3_42 + b_2_2·a_1_1·a_3_4
  21. b_3_52 + b_2_23 + a_2_1·b_2_22
  22. a_3_4·b_3_5 + b_2_2·a_4_4 + a_2_1·b_2_22
  23. a_2_0·a_4_4
  24. b_2_2·a_1_1·a_3_4 + a_2_1·a_4_4
  25. a_4_4·a_3_4
  26. a_4_4·b_3_5 + b_2_22·a_3_4 + a_2_1·b_2_2·b_3_5 + a_2_1·b_2_2·a_3_4 + a_2_0·c_4_8·a_1_1
  27. a_4_42 + a_2_1·b_2_2·a_4_4

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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_3, a Duflot regular element of degree 2
    2. c_4_8, a Duflot regular element of degree 4
    3. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 5].
  • 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 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. b_2_2 → 0, an element of degree 2
  6. c_2_3 → c_1_02, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. b_3_5 → 0, an element of degree 3
  9. a_4_4 → 0, an element of degree 4
  10. c_4_8 → c_1_14, an element of degree 4

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

  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. b_2_2 → c_1_22, an element of degree 2
  6. c_2_3 → c_1_22 + c_1_02, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. b_3_5 → c_1_23, an element of degree 3
  9. a_4_4 → 0, an element of degree 4
  10. c_4_8 → c_1_12·c_1_22 + c_1_14, 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