Cohomology of group number 1009 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 6.
  • Its center has rank 4.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 6.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 6 and depth 4.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t2  +  1
    (t  +  1)2 · (t  −  1)6
  • The a-invariants are -∞,-∞,-∞,-∞,-6,-6,-6. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 11 minimal generators of maximal degree 3:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. c_1_3, a Duflot regular element of degree 1
  5. b_2_7, an element of degree 2
  6. b_2_8, an element of degree 2
  7. b_2_9, an element of degree 2
  8. c_2_10, a Duflot regular element of degree 2
  9. c_2_11, a Duflot regular element of degree 2
  10. c_2_12, a Duflot regular element of degree 2
  11. b_3_31, an element of degree 3

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Ring relations

There are 18 minimal relations of maximal degree 6:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_2
  4. b_2_7·a_1_0
  5. b_2_8·a_1_0
  6. b_2_8·b_1_1 + b_2_7·b_1_2
  7. b_2_9·a_1_0
  8. b_2_72 + c_2_10·b_1_12
  9. b_2_82 + c_2_10·b_1_22
  10. b_2_7·b_2_8 + c_2_10·b_1_1·b_1_2
  11. b_2_9·b_1_1·b_1_2 + b_2_92 + c_2_12·b_1_12 + c_2_11·b_1_22
  12. b_1_2·b_3_31 + b_2_8·b_2_9
  13. a_1_0·b_3_31
  14. b_1_1·b_3_31 + b_2_7·b_2_9
  15. b_2_9·b_3_31 + b_2_7·b_2_9·b_1_2 + b_2_8·c_2_11·b_1_2 + b_2_7·c_2_12·b_1_1
  16. b_2_8·b_3_31 + b_2_9·c_2_10·b_1_2
  17. b_2_7·b_3_31 + b_2_9·c_2_10·b_1_1
  18. b_3_312 + b_2_92·c_2_10

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 6.
  • 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_1_3, a Duflot regular element of degree 1
    2. c_2_10, a Duflot regular element of degree 2
    3. c_2_11, a Duflot regular element of degree 2
    4. c_2_12, a Duflot regular element of degree 2
    5. b_1_22 + b_1_1·b_1_2 + b_1_12, an element of degree 2
    6. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 1, 3, 5].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -6, -6].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. c_1_3 → c_1_0, an element of degree 1
  5. b_2_7 → 0, an element of degree 2
  6. b_2_8 → 0, an element of degree 2
  7. b_2_9 → 0, an element of degree 2
  8. c_2_10 → c_1_32, an element of degree 2
  9. c_2_11 → c_1_12, an element of degree 2
  10. c_2_12 → c_1_22, an element of degree 2
  11. b_3_31 → 0, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 6

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_4, an element of degree 1
  3. b_1_2 → c_1_5, an element of degree 1
  4. c_1_3 → c_1_0, an element of degree 1
  5. b_2_7 → c_1_3·c_1_4, an element of degree 2
  6. b_2_8 → c_1_3·c_1_5, an element of degree 2
  7. b_2_9 → c_1_2·c_1_4 + c_1_1·c_1_5, an element of degree 2
  8. c_2_10 → c_1_32, an element of degree 2
  9. c_2_11 → c_1_1·c_1_4 + c_1_12, an element of degree 2
  10. c_2_12 → c_1_2·c_1_5 + c_1_22, an element of degree 2
  11. b_3_31 → c_1_2·c_1_3·c_1_4 + c_1_1·c_1_3·c_1_5, an element of degree 3

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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