Cohomology of group number 134 of order 128

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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 3.
  • Its center has rank 1.
  • It has 3 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 exceeds the Duflot bound, which is 1.
  • The Poincaré series is
     − 1
    (t  +  1) · (t  −  1)3
  • The a-invariants are -∞,-∞,-3,-3. 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. b_1_1, an element of degree 1
  3. b_2_1, an element of degree 2
  4. b_2_2, an element of degree 2
  5. b_2_3, an element of degree 2
  6. b_3_4, an element of degree 3
  7. b_3_5, an element of degree 3
  8. c_4_8, a Duflot regular element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_2_1·b_1_1
  4. b_2_2·a_1_0
  5. b_2_3·b_1_1 + b_2_3·a_1_0
  6. b_2_1·b_2_2
  7. a_1_0·b_3_4
  8. a_1_0·b_3_5
  9. b_1_1·b_3_5
  10. b_2_1·b_3_4
  11. b_2_3·b_3_4 + b_2_2·b_3_5
  12. b_3_42 + b_2_2·b_1_1·b_3_4 + b_2_23
  13. b_3_4·b_3_5 + b_2_22·b_2_3
  14. b_3_52 + b_2_2·b_2_32 + b_2_1·b_2_32

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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_4_8, a Duflot regular element of degree 4
    2. b_1_12 + b_2_3 + b_2_2 + b_2_1, an element of degree 2
    3. b_3_5 + b_2_2·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 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. b_1_1 → 0, an element of degree 1
  3. b_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_3_4 → 0, an element of degree 3
  7. b_3_5 → 0, an element of degree 3
  8. c_4_8 → c_1_04, an element of degree 4

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. b_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_3_4 → c_1_23 + c_1_12·c_1_2, an element of degree 3
  7. b_3_5 → 0, an element of degree 3
  8. c_4_8 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + c_1_04, an element of degree 4

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_12, an element of degree 2
  5. b_2_3 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_4 → c_1_13, an element of degree 3
  7. b_3_5 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. c_4_8 → c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2
       + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_04, an element of degree 4

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_2_1 → c_1_12, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_4 → 0, an element of degree 3
  7. b_3_5 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. c_4_8 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + 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