Cohomology of group number 7 of order 32

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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 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t2  −  t  +  1)
    (t  −  1)3 · (t2  +  1)
  • The a-invariants are -∞,-3,-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_3_3, an element of degree 3
  6. b_3_4, an element of degree 3
  7. b_4_6, an element of degree 4
  8. c_4_7, a Duflot regular element of degree 4

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Ring relations

There are 18 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_2_1·a_1_0
  4. b_2_2·a_1_0
  5. b_2_1·b_1_12 + b_2_12
  6. a_1_0·b_3_3
  7. b_1_1·b_3_3 + b_2_1·b_2_2
  8. a_1_0·b_3_4
  9. b_2_1·b_3_3 + b_2_1·b_2_2·b_1_1
  10. b_4_6·a_1_0
  11. b_4_6·b_1_1 + b_2_22·b_1_1 + b_2_1·b_3_4
  12. b_3_32 + b_2_1·b_2_22
  13. b_3_42 + b_2_2·b_1_1·b_3_4 + b_2_23 + b_2_1·b_1_1·b_3_4 + c_4_7·b_1_12
  14. b_2_1·b_1_1·b_3_4 + b_2_1·b_4_6 + b_2_1·b_2_22
  15. b_3_3·b_3_4 + b_2_2·b_4_6 + b_2_23
  16. b_4_6·b_3_3 + b_2_22·b_3_3 + b_2_1·b_2_2·b_3_4
  17. b_4_6·b_3_4 + b_2_22·b_3_4 + b_2_22·b_3_3 + b_2_1·b_2_2·b_3_4 + b_2_12·b_3_4
       + b_2_1·c_4_7·b_1_1
  18. b_4_62 + b_2_24 + b_2_1·b_2_2·b_4_6 + b_2_12·b_4_6 + b_2_12·b_2_22 + b_2_12·c_4_7

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_7, a Duflot regular element of degree 4
    2. b_1_12 + b_2_2, an element of degree 2
    3. b_1_1, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, 1, 3, 4].
  • 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 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_3_3 → 0, an element of degree 3
  6. b_3_4 → 0, an element of degree 3
  7. b_4_6 → 0, an element of degree 4
  8. c_4_7 → 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_3_3 → 0, an element of degree 3
  6. b_3_4 → c_1_23 + c_1_12·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  7. b_4_6 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  8. c_4_7 → 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 → c_1_2, an element of degree 1
  3. b_2_1 → c_1_22, an element of degree 2
  4. b_2_2 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  5. b_3_3 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  6. b_3_4 → c_1_23 + c_1_1·c_1_22 + c_1_13 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. b_4_6 → c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_14 + c_1_0·c_1_23
       + c_1_02·c_1_22, an element of degree 4
  8. c_4_7 → c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_0·c_1_23 + c_1_0·c_1_1·c_1_22
       + c_1_0·c_1_12·c_1_2 + 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