David J. Green

Cohomology
→Theory
→Implementation

Jena:

Faculty

External links:

Singular

Gap

Cohomology of group number 1024 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 4.
• It is non-abelian.
• It has p-Rank 5.
• Its center has rank 4.
• 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 coincides with the Duflot bound.
• 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.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring generators

The cohomology ring has 8 minimal generators of maximal degree 2:

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. c_1_3, a Duflot regular element of degree 1
5. b_2_7, an element of degree 2
6. c_2_8, a Duflot regular element of degree 2
7. c_2_9, a Duflot regular element of degree 2
8. c_2_10, a Duflot regular element of degree 2

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 5 minimal relations of maximal degree 4:

1. a_1_0²
2. a_1_0·b_1_1
3. b_1_2² + b_1_1·b_1_2 + a_1_0·b_1_2
4. b_2_7·a_1_0
5. b_1_1⁴ + b_2_7² + c_2_9·b_1_1²

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 5.
• However, the last relation was already found in degree 4 and the last generator in degree 2.
• The following is a filter regular homogeneous system of parameters:
  1. c_1_3, a Duflot regular element of degree 1
  2. c_2_8, a Duflot regular element of degree 2
  3. c_2_9, a Duflot regular element of degree 2
  4. c_2_10, a Duflot regular element of degree 2
  5. b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 2, 4].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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 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 → 0, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. b_2_7 → 0, an element of degree 2
6. c_2_8 → c_1_3² + c_1_1², an element of degree 2
7. c_2_9 → c_1_3², an element of degree 2
8. c_2_10 → c_1_2², an element of degree 2

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_4, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. b_2_7 → c_1_3·c_1_4, an element of degree 2
6. c_2_8 → c_1_3·c_1_4 + c_1_3² + c_1_1·c_1_4 + c_1_1², an element of degree 2
7. c_2_9 → c_1_4² + c_1_3², an element of degree 2
8. c_2_10 → c_1_2·c_1_4 + c_1_2², an element of degree 2

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_4, an element of degree 1
3. b_1_2 → c_1_4, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. b_2_7 → c_1_3·c_1_4, an element of degree 2
6. c_2_8 → c_1_3·c_1_4 + c_1_3² + c_1_1·c_1_4 + c_1_1², an element of degree 2
7. c_2_9 → c_1_4² + c_1_3², an element of degree 2
8. c_2_10 → c_1_4² + c_1_2·c_1_4 + c_1_2², an element of degree 2

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128