Cohomology of group number 84 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 4.
  • Its center has rank 2.
  • 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 exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1) · (t4  −  t3  −  1)
    (t  +  1)2 · (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 9 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_2_1, an element of degree 2
  4. b_2_2, an element of degree 2
  5. c_2_3, a Duflot regular element of degree 2
  6. a_3_4, a nilpotent element of degree 3
  7. b_3_5, an element of degree 3
  8. b_3_6, an element of degree 3
  9. c_4_10, a Duflot regular element of degree 4

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Ring relations

There are 16 minimal relations of maximal degree 6:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. b_2_1·a_1_1
  5. b_2_2·a_1_0
  6. a_1_0·a_3_4
  7. a_1_1·b_3_5 + a_1_1·a_3_4 + b_2_2·a_1_12
  8. a_1_0·b_3_5
  9. a_1_1·b_3_6 + a_1_1·a_3_4
  10. b_2_1·a_3_4
  11. a_1_12·a_3_4
  12. a_3_4·b_3_5 + a_3_42 + b_2_2·a_1_1·a_3_4 + b_2_22·a_1_12
  13. b_3_52 + b_2_1·b_2_22 + a_3_42
  14. a_3_4·b_3_6 + a_3_42
  15. b_3_62 + a_3_42 + b_2_12·c_2_3
  16. a_3_42 + b_2_2·a_1_1·a_3_4 + c_4_10·a_1_12 + b_2_2·c_2_3·a_1_12 + c_2_32·a_1_12

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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_2_3, a Duflot regular element of degree 2
    2. c_4_10, a Duflot regular element of degree 4
    3. b_2_2 + b_2_1, an element of degree 2
    4. b_3_5, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms 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 2

  1. a_1_0 → 0, an element of degree 1
  2. a_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. c_2_3 → c_1_02, an element of degree 2
  6. a_3_4 → 0, an element of degree 3
  7. b_3_5 → 0, an element of degree 3
  8. b_3_6 → 0, an element of degree 3
  9. c_4_10 → c_1_14, 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_1 → 0, an element of degree 1
  3. b_2_1 → c_1_22, an element of degree 2
  4. b_2_2 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  5. c_2_3 → c_1_32 + c_1_02, an element of degree 2
  6. a_3_4 → 0, an element of degree 3
  7. b_3_5 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  8. b_3_6 → c_1_22·c_1_3 + c_1_0·c_1_22, an element of degree 3
  9. c_4_10 → c_1_34 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, 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