Cohomology of group number 1166 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 4.
  • Its center has rank 3.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t2  +  t  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. a_1_1, a nilpotent element of degree 1
  4. b_1_3, an element of degree 1
  5. b_2_7, an element of degree 2
  6. c_2_8, a Duflot regular element of degree 2
  7. c_2_9, a Duflot regular element of degree 2
  8. c_4_31, a Duflot regular element of degree 4

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Ring relations

There are 6 minimal relations of maximal degree 4:

  1. a_1_22 + a_1_0·a_1_2
  2. a_1_12 + a_1_0·a_1_1 + a_1_02
  3. a_1_2·b_1_3 + a_1_2·a_1_1
  4. a_1_03
  5. b_2_7·a_1_2
  6. b_2_72 + b_2_7·a_1_0·b_1_3 + b_2_7·a_1_0·a_1_1 + c_2_9·b_1_32 + c_2_9·a_1_0·a_1_1
       + c_2_9·a_1_02 + c_2_8·a_1_0·a_1_2 + c_2_8·a_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 6.
  • However, the last relation was already found in degree 4 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_8, a Duflot regular element of degree 2
    2. c_2_9, a Duflot regular element of degree 2
    3. c_4_31, a Duflot regular element of degree 4
    4. b_1_32, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. a_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. b_2_7 → 0, an element of degree 2
  6. c_2_8 → c_1_22, an element of degree 2
  7. c_2_9 → c_1_12, an element of degree 2
  8. c_4_31 → c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. a_1_1 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_2_7 → c_1_1·c_1_3, an element of degree 2
  6. c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  7. c_2_9 → c_1_12, an element of degree 2
  8. c_4_31 → c_1_34 + c_1_04, 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