Cohomology of group number 31 of order 32

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General information on the group

  • The group has 3 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
     − 1

    (t  −  1)3 · (t2  +  1)
  • 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 6 minimal generators of maximal degree 4:

  1. a_1_1, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_2, an element of degree 1
  4. c_2_4, a Duflot regular element of degree 2
  5. b_3_6, an element of degree 3
  6. c_4_9, a Duflot regular element of degree 4

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Ring relations

There are 5 minimal relations of maximal degree 6:

  1. a_1_1·b_1_0 + a_1_12
  2. b_1_22 + b_1_0·b_1_2 + a_1_12
  3. a_1_13
  4. a_1_1·b_3_6 + c_2_4·a_1_12
  5. b_3_62 + b_1_02·b_1_2·b_3_6 + b_1_05·b_1_2 + c_4_9·b_1_02 + c_2_4·b_1_03·b_1_2
       + c_2_4·b_1_04 + c_4_9·a_1_12 + c_2_42·b_1_02


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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_4, a Duflot regular element of degree 2
    2. c_4_9, a Duflot regular element of degree 4
    3. b_1_0, 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 2

  1. a_1_10, an element of degree 1
  2. b_1_00, an element of degree 1
  3. b_1_20, an element of degree 1
  4. c_2_4c_1_12 + c_1_02, an element of degree 2
  5. b_3_60, an element of degree 3
  6. c_4_9c_1_04, an element of degree 4

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

  1. a_1_10, an element of degree 1
  2. b_1_0c_1_2, an element of degree 1
  3. b_1_20, an element of degree 1
  4. c_2_4c_1_1·c_1_2 + c_1_12 + c_1_0·c_1_2 + c_1_02, an element of degree 2
  5. b_3_6c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  6. c_4_9c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_04, an element of degree 4

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

  1. a_1_10, an element of degree 1
  2. b_1_0c_1_2, an element of degree 1
  3. b_1_2c_1_2, an element of degree 1
  4. c_2_4c_1_1·c_1_2 + c_1_12 + c_1_0·c_1_2 + c_1_02, an element of degree 2
  5. b_3_6c_1_23 + c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  6. c_4_9c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_02·c_1_22 + 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