Cohomology of group number 1613 of order 128

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General information on the group

  • The group has 4 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) · (t  −  1)5
  • 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 10 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. c_1_2, a Duflot regular element of degree 1
  4. c_1_3, a Duflot regular element of degree 1
  5. b_2_8, an element of degree 2
  6. b_2_9, an element of degree 2
  7. b_2_10, an element of degree 2
  8. b_3_22, an element of degree 3
  9. b_3_23, an element of degree 3
  10. c_4_45, 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_8·b_1_1 + b_2_9·a_1_0
  4. b_2_9·b_1_1 + b_2_8·b_1_1
  5. b_2_8·b_1_1 + b_2_10·a_1_0
  6. b_2_92 + b_2_8·b_2_10
  7. a_1_0·b_3_22
  8. b_1_1·b_3_22
  9. a_1_0·b_3_23
  10. b_2_10·b_3_22 + b_2_9·b_3_23 + b_2_8·b_2_9·a_1_0
  11. b_2_9·b_3_22 + b_2_8·b_3_23 + b_2_8·b_2_9·a_1_0
  12. b_3_222 + b_2_8·b_2_102 + b_2_82·b_2_10
  13. b_3_22·b_3_23 + b_2_9·b_2_102 + b_2_8·b_2_9·b_2_10
  14. b_3_232 + b_2_10·b_1_1·b_3_23 + b_2_103 + b_2_8·b_2_102 + c_4_45·b_1_12

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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_2, a Duflot regular element of degree 1
    2. c_1_3, a Duflot regular element of degree 1
    3. c_4_45, a Duflot regular element of degree 4
    4. b_1_12 + b_2_10 + b_2_9 + b_2_8, an element of degree 2
    5. b_3_22 + b_2_10·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 3, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
  • We found that there exists some filter regular HSOP formed by the first 3 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 3

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. c_1_3 → c_1_1, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_2_10 → 0, an element of degree 2
  8. b_3_22 → 0, an element of degree 3
  9. b_3_23 → 0, an element of degree 3
  10. c_4_45 → c_1_24, 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. c_1_2 → c_1_0, an element of degree 1
  4. c_1_3 → c_1_1, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_2_10 → c_1_42 + c_1_3·c_1_4, an element of degree 2
  8. b_3_22 → 0, an element of degree 3
  9. b_3_23 → c_1_43 + c_1_32·c_1_4 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  10. c_4_45 → c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4 + c_1_22·c_1_42 + c_1_22·c_1_3·c_1_4
       + c_1_22·c_1_32 + c_1_24, 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. c_1_2 → c_1_0, an element of degree 1
  4. c_1_3 → c_1_1, an element of degree 1
  5. b_2_8 → c_1_42, an element of degree 2
  6. b_2_9 → c_1_3·c_1_4, an element of degree 2
  7. b_2_10 → c_1_32, an element of degree 2
  8. b_3_22 → c_1_3·c_1_42 + c_1_32·c_1_4, an element of degree 3
  9. b_3_23 → c_1_32·c_1_4 + c_1_33, an element of degree 3
  10. c_4_45 → c_1_3·c_1_43 + c_1_33·c_1_4 + c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4
       + c_1_22·c_1_42 + c_1_22·c_1_3·c_1_4 + c_1_22·c_1_32 + c_1_24, 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