Cohomology of group number 2106 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 2.
  • Its center has rank 2.
  • 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t8  +  2·t7  +  3·t6  +  2·t5  +  t4  +  2·t3  +  3·t2  +  2·t  +  1
    (t  −  1)2 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-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_1_2, a nilpotent element of degree 1
  4. a_1_3, a nilpotent element of degree 1
  5. c_4_8, a Duflot regular element of degree 4
  6. a_5_7, a nilpotent element of degree 5
  7. a_5_8, a nilpotent element of degree 5
  8. a_5_9, a nilpotent element of degree 5
  9. c_8_17, a Duflot regular element of degree 8

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Ring relations

There are 17 minimal relations of maximal degree 10:

  1. a_1_0·a_1_2
  2. a_1_32 + a_1_1·a_1_3 + a_1_12 + a_1_0·a_1_1
  3. a_1_13 + a_1_02·a_1_1
  4. a_1_23 + a_1_12·a_1_2 + a_1_03
  5. a_1_03·a_1_12
  6. a_1_2·a_5_7
  7. a_1_0·a_5_8 + c_4_8·a_1_0·a_1_1 + c_4_8·a_1_02
  8. a_1_1·a_5_7 + a_1_0·a_5_9 + a_1_0·a_5_7 + c_4_8·a_1_0·a_1_3 + c_4_8·a_1_0·a_1_1
       + c_4_8·a_1_02
  9. a_1_2·a_5_9 + a_1_2·a_5_8 + a_1_1·a_5_8 + c_4_8·a_1_2·a_1_3 + c_4_8·a_1_22
       + c_4_8·a_1_12 + c_4_8·a_1_0·a_1_1
  10. a_1_22·a_5_8 + a_1_12·a_5_8 + a_1_02·a_5_7 + c_4_8·a_1_1·a_1_22
       + c_4_8·a_1_0·a_1_12 + c_4_8·a_1_02·a_1_1
  11. a_1_12·a_5_9 + a_1_12·a_5_8 + a_1_0·a_1_1·a_5_9 + c_4_8·a_1_12·a_1_3
       + c_4_8·a_1_12·a_1_2 + c_4_8·a_1_0·a_1_1·a_1_3 + c_4_8·a_1_0·a_1_12
       + c_4_8·a_1_02·a_1_1
  12. a_5_7·a_5_8 + a_1_03·a_1_1·a_1_3·a_5_9 + c_4_8·a_1_0·a_5_9 + c_4_82·a_1_0·a_1_3
       + c_4_82·a_1_0·a_1_1 + c_4_82·a_1_02
  13. a_5_92 + a_5_82 + a_5_72 + c_8_17·a_1_12 + c_4_82·a_1_22 + c_4_82·a_1_1·a_1_3
       + c_4_82·a_1_12 + c_4_82·a_1_0·a_1_1
  14. a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_72 + c_8_17·a_1_0·a_1_1 + c_4_8·a_1_3·a_5_7
  15. a_5_72 + a_1_03·a_1_1·a_1_3·a_5_9 + c_8_17·a_1_02
  16. a_5_8·a_5_9 + a_5_82 + a_5_7·a_5_8 + a_1_03·a_1_1·a_1_3·a_5_9 + c_8_17·a_1_1·a_1_2
       + c_4_8·a_1_3·a_5_8 + c_4_8·a_1_2·a_5_8 + c_4_8·a_1_1·a_5_9 + c_4_8·a_1_1·a_5_8
       + c_4_82·a_1_1·a_1_3 + c_4_82·a_1_1·a_1_2
  17. a_5_82 + c_8_17·a_1_22 + c_4_82·a_1_12 + c_4_82·a_1_02

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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_8_17, a Duflot regular element of degree 8
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 10].
  • 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 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. a_1_3 → 0, an element of degree 1
  5. c_4_8 → c_1_04, an element of degree 4
  6. a_5_7 → 0, an element of degree 5
  7. a_5_8 → 0, an element of degree 5
  8. a_5_9 → 0, an element of degree 5
  9. c_8_17 → c_1_18, 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