Cohomology of group number 8 of order 32

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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 2.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t5  +  t2  +  1
    (t  −  1)2 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_2_2, an element of degree 2
  5. a_3_3, a nilpotent element of degree 3
  6. a_5_2, a nilpotent element of degree 5
  7. b_5_4, an element of degree 5
  8. a_6_4, a nilpotent element of degree 6
  9. c_8_6, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_1·a_1_0
  4. b_2_2·a_1_0 + a_1_13
  5. a_2_12 + a_2_1·a_1_12
  6. a_2_1·b_2_2 + a_1_1·a_3_3
  7. a_1_0·a_3_3 + a_2_12
  8. b_2_2·a_1_13
  9. a_2_1·a_3_3 + a_1_12·a_3_3
  10. a_3_32 + b_2_2·a_1_1·a_3_3 + b_2_22·a_1_12 + a_1_13·a_3_3
  11. a_1_1·a_5_2
  12. a_3_32 + a_1_0·a_5_2 + b_2_2·a_1_1·a_3_3 + b_2_22·a_1_12
  13. a_1_0·b_5_4 + a_3_32 + b_2_2·a_1_1·a_3_3 + b_2_22·a_1_12
  14. a_2_1·a_5_2
  15. b_2_2·a_5_2 + a_1_12·b_5_4
  16. a_2_1·b_5_4 + a_6_4·a_1_1 + b_2_2·a_1_12·a_3_3
  17. a_6_4·a_1_0
  18. a_3_3·b_5_4 + b_2_2·a_6_4 + a_3_3·a_5_2 + b_2_22·a_1_1·a_3_3
  19. a_3_3·a_5_2 + a_2_1·a_6_4
  20. a_3_3·a_5_2 + a_6_4·a_1_12
  21. a_6_4·a_3_3 + b_2_2·a_1_12·b_5_4 + b_2_2·a_6_4·a_1_1
  22. a_5_22
  23. a_5_2·b_5_4 + b_2_24·a_1_12
  24. b_5_42 + b_2_25 + b_2_22·a_1_1·b_5_4 + b_2_24·a_1_12 + c_8_6·a_1_12
  25. a_6_4·a_5_2 + b_2_23·a_1_12·a_3_3
  26. a_6_4·b_5_4 + b_2_24·a_3_3 + a_2_1·c_8_6·a_1_1
  27. a_6_42 + b_2_24·a_1_1·a_3_3 + b_2_25·a_1_12 + b_2_22·a_6_4·a_1_12
       + a_2_1·c_8_6·a_1_12

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_6, a Duflot regular element of degree 8
    2. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 6, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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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_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. a_5_2 → 0, an element of degree 5
  7. b_5_4 → 0, an element of degree 5
  8. a_6_4 → 0, an element of degree 6
  9. c_8_6 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. 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_1 → 0, an element of degree 2
  4. b_2_2 → c_1_12, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. a_5_2 → 0, an element of degree 5
  7. b_5_4 → c_1_15, an element of degree 5
  8. a_6_4 → 0, an element of degree 6
  9. c_8_6 → c_1_18 + 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