Cohomology of group number 2157 of order 128

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General information on the group

  • The group has 5 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its center has rank 3.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 5.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 5 and depth 4.
  • The depth exceeds the Duflot bound, which is 3.
  • The Poincaré series is
     − 1
    (t  −  1)5 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-∞,-5,-5. 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_1_2, an element of degree 1
  4. b_1_3, an element of degree 1
  5. c_1_4, a Duflot regular element of degree 1
  6. c_2_13, a Duflot regular element of degree 2
  7. b_3_29, an element of degree 3
  8. c_4_55, 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_02
  2. b_1_1·b_1_2 + a_1_0·b_1_3 + a_1_0·b_1_1
  3. a_1_0·b_1_32 + a_1_0·b_1_2·b_1_3 + a_1_0·b_1_1·b_1_3
  4. a_1_0·b_3_29
  5. b_3_292 + c_2_13·b_1_34 + c_2_13·b_1_22·b_1_32 + c_2_13·b_1_12·b_1_32

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 7.
  • However, the last relation was already found in degree 6 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_1_4, a Duflot regular element of degree 1
    2. c_2_13, a Duflot regular element of degree 2
    3. c_4_55, a Duflot regular element of degree 4
    4. b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_3 + b_1_12, an element of degree 2
    5. b_1_32, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 4, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. c_2_13 → c_1_22, an element of degree 2
  7. b_3_29 → 0, an element of degree 3
  8. c_4_55 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_4, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. c_2_13 → c_1_22, an element of degree 2
  7. b_3_29 → c_1_2·c_1_42 + c_1_2·c_1_3·c_1_4, an element of degree 3
  8. c_4_55 → c_1_2·c_1_43 + c_1_2·c_1_32·c_1_4 + c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4
       + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_1_3 → c_1_4, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. c_2_13 → c_1_22, an element of degree 2
  7. b_3_29 → c_1_2·c_1_42 + c_1_2·c_1_3·c_1_4, an element of degree 3
  8. c_4_55 → c_1_2·c_1_43 + c_1_2·c_1_32·c_1_4 + c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4
       + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32 + 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