Cohomology of group number 2084 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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) · (t5  −  2·t4  +  2·t2  +  2·t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-4,-3. 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 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, an element of degree 1
  4. b_1_3, an element of degree 1
  5. c_4_8, a Duflot regular element of degree 4
  6. c_4_9, a Duflot regular element of degree 4
  7. a_5_5, a nilpotent element of degree 5
  8. a_5_12, a nilpotent element of degree 5
  9. a_5_7, a nilpotent element of degree 5

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Ring relations

There are 18 minimal relations of maximal degree 10:

  1. a_1_0·a_1_2
  2. b_1_1·b_1_3 + a_1_0·b_1_1 + a_1_22 + a_1_02
  3. a_1_2·b_1_12 + a_1_0·b_1_32 + a_1_03
  4. a_1_22·b_1_3 + a_1_22·b_1_1 + a_1_02·b_1_3 + a_1_02·b_1_1 + a_1_03
  5. a_1_0·a_5_5 + c_4_9·a_1_02
  6. a_1_2·a_5_5 + c_4_8·a_1_22
  7. b_1_3·a_5_12 + b_1_1·a_5_5 + a_1_0·b_1_15 + c_4_9·a_1_2·b_1_3 + c_4_9·a_1_0·b_1_3
       + c_4_9·a_1_0·b_1_1 + c_4_8·a_1_2·b_1_1 + c_4_8·a_1_0·b_1_3
  8. b_1_1·a_5_5 + a_1_0·b_1_15 + a_1_0·a_5_12 + c_4_9·a_1_0·b_1_1 + c_4_8·a_1_2·b_1_1
       + c_4_9·a_1_02 + c_4_8·a_1_02
  9. a_1_2·a_5_12 + c_4_9·a_1_22
  10. b_1_1·a_5_7 + b_1_1·a_5_5 + a_1_0·b_1_15 + c_4_9·a_1_2·b_1_1 + c_4_9·a_1_0·b_1_1
       + c_4_8·a_1_0·b_1_1
  11. a_1_0·a_5_7 + c_4_8·a_1_02
  12. b_1_3·a_5_5 + a_1_2·a_5_7 + c_4_9·a_1_0·b_1_3 + c_4_8·a_1_2·b_1_3 + c_4_9·a_1_22
       + c_4_8·a_1_22
  13. a_5_52 + c_4_92·a_1_02 + c_4_82·a_1_22
  14. a_5_5·a_5_12 + a_1_0·b_1_14·a_5_12 + c_4_9·a_1_0·a_5_12 + c_4_8·c_4_9·a_1_22
  15. a_5_122 + a_1_0·b_1_14·a_5_12 + c_4_92·a_1_22 + c_4_92·a_1_02
       + c_4_82·a_1_02
  16. a_5_72 + a_1_2·b_1_34·a_5_7 + c_4_92·a_1_22 + c_4_82·a_1_22 + c_4_82·a_1_02
  17. a_5_5·a_5_7 + c_4_8·a_1_2·a_5_7 + c_4_8·c_4_9·a_1_02
  18. a_5_12·a_5_7 + c_4_9·a_1_2·a_5_7 + c_4_8·a_1_0·a_5_12

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 10.
  • 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_4_8, a Duflot regular element of degree 4
    2. c_4_9, a Duflot regular element of degree 4
    3. b_1_32 + b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 7].
  • 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_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_4_8 → c_1_04, an element of degree 4
  6. c_4_9 → c_1_14, an element of degree 4
  7. a_5_5 → 0, an element of degree 5
  8. a_5_12 → 0, an element of degree 5
  9. a_5_7 → 0, an element of degree 5

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_4_8 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  6. c_4_9 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  7. a_5_5 → 0, an element of degree 5
  8. a_5_12 → 0, an element of degree 5
  9. a_5_7 → 0, an element of degree 5

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. c_4_8 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  6. c_4_9 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  7. a_5_5 → 0, an element of degree 5
  8. a_5_12 → 0, an element of degree 5
  9. a_5_7 → 0, an element of degree 5

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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