Cohomology of group number 6671 of order 256

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General information on the group

  • The group is the Sylow 2-subgroup of SL(4,3).
  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its centre 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 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
     − 1  −  2·t2  +  t3  −  2·t4  +  2·t5  −  2·t6  +  t7
    ( − 1  +  t)3 · (1  +  t2)2 · (1  +  t4)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Symonds test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Ring generators

The cohomology ring has 11 minimal generators of maximal degree 8:

  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. a_2_4, a nilpotent element of degree 2
  5. a_2_5, a nilpotent element of degree 2
  6. b_3_8, an element of degree 3
  7. b_4_11, an element of degree 4
  8. a_5_12, a nilpotent element of degree 5
  9. b_6_17, an element of degree 6
  10. b_7_21, an element of degree 7
  11. c_8_25, a Duflot element of degree 8

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Ring relations

There are 35 minimal relations of maximal degree 14:

  1. a_1_0·b_1_1
  2. a_1_0·b_1_2
  3. a_2_4·b_1_2
  4. a_2_5·a_1_0 + a_1_03
  5. a_2_5·b_1_1 + a_2_4·b_1_1
  6. a_1_04
  7. a_2_4·a_2_5 + a_2_42
  8. a_1_0·b_3_8
  9. b_1_2·b_3_8 + b_1_1·b_3_8
  10. a_2_5·b_3_8 + a_2_4·a_1_03
  11. a_2_4·b_3_8 + a_2_4·a_1_03
  12. b_4_11·b_1_2 + b_4_11·b_1_1
  13. a_2_43
  14. a_2_53
  15. b_3_82 + b_4_11·b_1_12
  16. b_1_2·a_5_12 + a_2_5·b_1_24 + a_2_52·b_1_22
  17. b_1_1·a_5_12 + a_2_4·b_1_14
  18. a_2_5·a_5_12 + a_2_52·b_1_23 + a_2_42·b_1_13 + a_1_02·a_5_12 + b_4_11·a_1_03
  19. b_6_17·a_1_0 + a_2_5·a_5_12 + a_2_52·b_1_23 + a_2_4·b_4_11·a_1_0 + a_2_42·b_1_13
  20. a_2_52·b_4_11 + a_1_03·a_5_12
  21. b_3_8·a_5_12 + a_2_42·b_4_11 + a_2_4·b_4_11·a_1_02
  22. a_2_5·b_6_17 + a_2_52·b_1_24 + a_2_52·b_4_11 + a_2_42·b_1_14 + a_2_42·b_4_11
  23. a_2_4·b_6_17 + a_2_42·b_1_14 + a_2_42·b_4_11 + a_2_4·a_1_0·a_5_12
       + a_2_4·b_4_11·a_1_02
  24. a_1_0·b_7_21 + a_2_42·b_4_11 + a_2_4·a_1_0·a_5_12
  25. b_1_2·b_7_21 + b_1_1·b_7_21 + b_6_17·b_1_22 + b_6_17·b_1_1·b_1_2 + a_2_5·b_1_26
       + a_2_4·b_1_16 + a_2_42·b_1_14
  26. a_2_5·b_7_21 + a_2_42·b_1_15
  27. a_2_4·b_7_21 + a_2_42·b_1_15 + a_2_4·b_4_11·a_1_03
  28. b_1_12·b_7_21 + b_1_16·b_3_8 + b_6_17·b_3_8 + b_6_17·b_1_12·b_1_2 + b_4_11·b_1_15
       + a_2_4·b_1_17 + a_2_42·b_1_15 + a_2_4·b_4_11·a_1_03
  29. b_3_8·b_7_21 + b_6_17·b_1_1·b_3_8 + b_4_11·b_1_13·b_3_8 + b_4_11·b_1_16
       + b_4_11·b_6_17 + a_2_4·b_4_112 + b_4_11·a_1_0·a_5_12 + b_4_112·a_1_02
  30. a_2_5·b_4_112 + a_5_122 + b_4_11·a_1_0·a_5_12 + b_4_112·a_1_02 + a_2_52·b_1_26
       + a_2_42·b_1_16 + c_8_25·a_1_02
  31. b_6_17·a_5_12 + a_2_52·b_1_27 + a_2_4·b_4_11·a_5_12 + a_2_42·b_1_17
       + c_8_25·a_1_03
  32. b_6_17·b_1_12·b_1_24 + b_6_172 + b_4_112·b_1_14 + a_5_12·b_7_21
       + a_2_52·b_1_28 + a_2_4·a_5_122 + a_2_4·b_4_11·a_1_0·a_5_12
       + c_8_25·b_1_12·b_1_22
  33. a_5_12·b_7_21 + a_2_42·b_1_18 + a_2_4·b_4_112·a_1_02 + b_4_11·a_1_03·a_5_12
       + a_2_4·c_8_25·a_1_02
  34. b_6_17·b_7_21 + b_6_172·b_1_2 + b_4_11·b_6_17·b_1_13 + b_4_112·b_1_12·b_3_8
       + a_2_52·b_1_29 + a_2_42·b_1_19 + a_2_4·b_4_11·a_1_02·a_5_12
       + a_2_4·b_4_112·a_1_03 + c_8_25·b_1_12·b_3_8 + a_2_4·c_8_25·a_1_03
  35. b_7_212 + b_6_172·b_1_22 + b_4_11·b_1_110 + b_4_11·b_6_17·b_1_14
       + b_4_112·b_1_16 + b_4_113·b_1_12 + a_2_52·b_1_210 + a_2_42·b_1_110
       + b_4_11·c_8_25·b_1_12

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Symonds test

  • We proved completion in degree 14 using the Symonds criterion.
  • 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_8_25, an element of degree 8
    2. b_1_24 + b_1_12·b_1_22 + b_1_14 + b_4_11, an element of degree 4
    3. b_1_1, an element of degree 1
  • A Duflot regular sequence is given by c_8_25.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 9, 10].
  • As a module over these parameters, the cohomology is generated in degree at most 10.

About the group Ring generators Ring relations Completion information Restriction maps

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_1_2 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_4_11 → 0, an element of degree 4
  8. a_5_12 → 0, an element of degree 5
  9. b_6_17 → 0, an element of degree 6
  10. b_7_21 → 0, an element of degree 7
  11. c_8_25 → c_1_08, an element of degree 8

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_1_2 → c_1_2, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_3_8 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_4_11 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  8. a_5_12 → 0, an element of degree 5
  9. b_6_17 → c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_24
       + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22, an element of degree 6
  10. b_7_21 → c_1_1·c_1_26 + c_1_14·c_1_23 + c_1_0·c_1_1·c_1_25 + c_1_0·c_1_14·c_1_22
       + c_1_02·c_1_25 + c_1_02·c_1_12·c_1_23 + c_1_02·c_1_14·c_1_2
       + c_1_04·c_1_23 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
  11. c_8_25 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25
       + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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_1_2 → c_1_2, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_4_11 → 0, an element of degree 4
  8. a_5_12 → 0, an element of degree 5
  9. b_6_17 → c_1_0·c_1_12·c_1_23 + c_1_0·c_1_13·c_1_22 + c_1_02·c_1_1·c_1_23
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_1·c_1_2, an element of degree 6
  10. b_7_21 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_13·c_1_23 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_12·c_1_23 + c_1_02·c_1_13·c_1_22 + c_1_04·c_1_1·c_1_22, an element of degree 7
  11. c_8_25 → c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24 + c_1_02·c_1_1·c_1_25
       + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010