David J. Green

Cohomology
→Theory
→Implementation

Jena:

Faculty

External links:

Singular

Gap

Cohomology of group number 1739 of order 128

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

General information on the group

• The group has 4 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t5  +  t4  −  t3  +  t2  +  1

  (t  −  1)4 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-6,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

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. b_1_3, an element of degree 1
5. b_2_8, an element of degree 2
6. c_2_9, a Duflot regular element of degree 2
7. a_5_16, a nilpotent element of degree 5
8. b_5_46, an element of degree 5
9. b_5_47, an element of degree 5
10. b_6_69, an element of degree 6
11. c_8_135, a Duflot regular element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 27 minimal relations of maximal degree 12:

1. a_1_0·b_1_1
2. a_1_0·b_1_2
3. b_1_1·b_1_3² + a_1_0³
4. b_2_8·a_1_0
5. b_1_2²·b_1_3² + b_2_8·b_1_1·b_1_2 + b_2_8² + c_2_9·b_1_1²
6. a_1_0³·b_1_3²
7. b_1_2·b_1_3⁵ + b_2_8·b_1_3⁴ + b_1_2·a_5_16
8. b_1_1·a_5_16
9. a_1_0·b_5_46
10. a_1_0·b_5_47
11. b_1_2·b_5_46 + b_1_1·b_5_47
12. b_1_2·b_1_3·a_5_16 + b_2_8·a_5_16
13. b_1_3²·b_5_46 + a_1_0²·b_1_3⁵ + a_1_0²·a_5_16
14. b_6_69·b_1_2 + b_2_8·b_5_47 + b_2_8²·b_1_3³
15. b_6_69·a_1_0
16. b_6_69·b_1_1 + b_2_8·b_5_46
17. b_1_2·b_1_3²·b_5_47 + b_2_8·b_1_1·b_5_47 + b_2_8·b_6_69 + b_2_8²·b_1_3⁴
       + b_2_8·b_1_2·a_5_16 + c_2_9·b_1_1·b_5_46
18. a_5_16·b_5_46 + a_1_0²·b_1_3³·a_5_16
19. b_1_3⁵·b_5_47 + b_6_69·b_1_3⁴ + b_2_8·b_1_3⁸ + a_5_16·b_5_47 + b_2_8·b_1_3³·a_5_16
20. b_5_47² + c_8_135·b_1_2²
21. b_5_46·b_5_47 + c_8_135·b_1_1·b_1_2
22. b_5_46² + c_8_135·b_1_1²
23. a_5_16² + a_1_0·b_1_3⁴·a_5_16 + a_1_0²·b_1_3⁸ + c_8_135·a_1_0²
24. b_1_3·a_5_16·b_5_47 + b_6_69·a_5_16 + b_2_8·b_1_3⁴·a_5_16
25. b_6_69·b_5_47 + b_2_8·b_6_69·b_1_3³ + b_2_8²·b_1_3⁷ + b_2_8²·b_1_3²·a_5_16
       + b_2_8·c_8_135·b_1_2
26. b_6_69·b_5_46 + b_2_8·c_8_135·b_1_1
27. b_6_69² + b_2_8²·b_1_3⁸ + b_2_8²·c_8_135

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Data used for Benson′s test

• Benson′s completion test succeeded in degree 12.
• 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_9, a Duflot regular element of degree 2
  2. c_8_135, a Duflot regular element of degree 8
  3. b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_1·b_1_2 + b_1_1², an element of degree 2
  4. b_1_2²·b_1_3 + b_1_1·b_1_2² + b_1_1²·b_1_2 + b_2_8·b_1_3, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 11].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

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. b_2_8 → 0, an element of degree 2
6. c_2_9 → c_1_0², an element of degree 2
7. a_5_16 → 0, an element of degree 5
8. b_5_46 → 0, an element of degree 5
9. b_5_47 → 0, an element of degree 5
10. b_6_69 → 0, an element of degree 6
11. c_8_135 → c_1_1⁸, an element of degree 8

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

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_3, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_2_8 → c_1_0·c_1_2, an element of degree 2
6. c_2_9 → c_1_0·c_1_3 + c_1_0², an element of degree 2
7. a_5_16 → 0, an element of degree 5
8. b_5_46 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
9. b_5_47 → c_1_1²·c_1_2²·c_1_3 + c_1_1⁴·c_1_3, an element of degree 5
10. b_6_69 → c_1_0·c_1_1²·c_1_2³ + c_1_0·c_1_1⁴·c_1_2, an element of degree 6
11. c_8_135 → c_1_1⁴·c_1_2⁴ + c_1_1⁸, an element of degree 8

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

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_2, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_2_8 → c_1_2·c_1_3, an element of degree 2
6. c_2_9 → c_1_0·c_1_2 + c_1_0², an element of degree 2
7. a_5_16 → 0, an element of degree 5
8. b_5_46 → 0, an element of degree 5
9. b_5_47 → c_1_1²·c_1_2·c_1_3² + c_1_1⁴·c_1_2, an element of degree 5
10. b_6_69 → c_1_2·c_1_3⁵ + c_1_1²·c_1_2·c_1_3³ + c_1_1⁴·c_1_2·c_1_3, an element of degree 6
11. c_8_135 → c_1_1⁴·c_1_3⁴ + c_1_1⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128