Cohomology of group number 928 of order 128

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General information on the group

  • The group is also known as Syl2(S8), the Sylow 2-subgroup of Symmetric Group S_8.
  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 1.
  • It has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 4, 4 and 4, respectively.

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 1.
  • The Poincaré series is
    1
    (t  +  1) · (t  −  1)4
  • 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. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. b_2_6, an element of degree 2
  7. b_3_11, an element of degree 3
  8. b_3_12, an element of degree 3
  9. c_4_21, a Duflot regular element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

  1. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. b_2_4·b_1_2
  4. b_2_5·b_1_1
  5. b_2_6·b_1_0
  6. b_2_4·b_2_5
  7. b_1_0·b_3_11
  8. b_1_0·b_3_12
  9. b_1_2·b_3_11 + b_1_1·b_3_12
  10. b_2_5·b_3_11
  11. b_2_4·b_3_12
  12. b_3_122 + b_2_6·b_1_2·b_3_12 + b_2_5·b_2_62 + c_4_21·b_1_22
  13. b_3_11·b_3_12 + b_2_6·b_1_1·b_3_12 + c_4_21·b_1_1·b_1_2
  14. b_3_112 + b_2_6·b_1_1·b_3_11 + b_2_4·b_2_62 + c_4_21·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_4_21, a Duflot regular element of degree 4
    2. b_1_2·b_3_12 + b_1_24 + b_1_1·b_3_11 + b_1_12·b_1_22 + b_1_14 + b_1_04
         + b_2_6·b_1_22 + b_2_6·b_1_1·b_1_2 + b_2_62 + b_2_52 + b_2_42, an element of degree 4
    3. b_1_23·b_3_12 + b_1_1·b_1_22·b_3_12 + b_1_12·b_1_24 + b_1_13·b_3_12
         + b_1_13·b_3_11 + b_1_14·b_1_22 + b_2_6·b_1_24 + b_2_6·b_1_13·b_1_2
         + b_2_62·b_1_22 + b_2_62·b_1_1·b_1_2 + b_2_62·b_1_12 + b_2_5·b_1_2·b_3_12
         + b_2_5·b_2_6·b_1_22 + b_2_5·b_2_62 + b_2_52·b_1_22 + b_2_52·b_1_02
         + b_2_4·b_1_1·b_3_11 + b_2_4·b_2_62 + b_2_42·b_1_12 + b_2_42·b_1_02, an element of degree 6
    4. b_1_1·b_1_23·b_3_12 + b_1_12·b_1_22·b_3_12 + b_1_13·b_1_2·b_3_12 + b_1_14·b_3_12
         + b_2_6·b_1_1·b_1_24 + b_2_6·b_1_13·b_1_22 + b_2_62·b_1_1·b_1_22
         + b_2_62·b_1_12·b_1_2 + b_2_5·b_1_22·b_3_12 + b_2_5·b_2_6·b_1_23
         + b_2_5·b_2_62·b_1_2 + b_2_4·b_1_12·b_3_11 + b_2_4·b_2_62·b_1_1, an element of degree 7
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 10, 17].
  • 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 term of the above HSOP, together with 3 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. b_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_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_2_6 → 0, an element of degree 2
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. c_4_21 → c_1_04, an element of degree 4

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

  1. b_1_0 → c_1_1, 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_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_2_6 → 0, an element of degree 2
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. c_4_21 → 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. b_1_0 → c_1_1, 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_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_2_6 → 0, an element of degree 2
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. c_4_21 → 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 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_2_6 → c_1_32 + c_1_2·c_1_3 + c_1_0·c_1_2 + c_1_0·c_1_1, an element of degree 2
  7. b_3_11 → c_1_1·c_1_32 + c_1_1·c_1_2·c_1_3 + c_1_0·c_1_1·c_1_2 + c_1_02·c_1_1, an element of degree 3
  8. b_3_12 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  9. c_4_21 → c_1_0·c_1_1·c_1_32 + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_2 + c_1_03·c_1_2 + c_1_03·c_1_1 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_2_6 → c_1_32 + c_1_2·c_1_3 + c_1_0·c_1_1, an element of degree 2
  7. b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_1·c_1_2·c_1_3 + c_1_02·c_1_1, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. c_4_21 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_1·c_1_32
       + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22
       + c_1_02·c_1_1·c_1_2 + c_1_03·c_1_1 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_1, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_2_6 → c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_0·c_1_1, an element of degree 2
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_12
       + c_1_02·c_1_1, an element of degree 3
  9. c_4_21 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_1·c_1_2·c_1_3
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_3
       + c_1_02·c_1_1·c_1_2 + c_1_03·c_1_1 + c_1_04, an element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


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