David J. Green

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Cohomology of group number 1070 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 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. b_1_3, an 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

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Ring relations

There are 5 minimal relations of maximal degree 4:

1. a_1_0²
2. a_1_0·b_1_2
3. b_1_1·b_1_3 + a_1_0·b_1_1
4. b_2_7·a_1_0
5. b_2_7² + c_2_8·b_1_2²

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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_2_8, a Duflot regular element of degree 2
  2. c_2_9, a Duflot regular element of degree 2
  3. c_2_10, a Duflot regular element of degree 2
  4. 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
  5. b_1_2², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 3, 5].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

About the group

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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. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_2_7 → 0, an element of degree 2
6. c_2_8 → c_1_2², an element of degree 2
7. c_2_9 → c_1_1², an element of degree 2
8. c_2_10 → c_1_0², 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_3, an element of degree 1
3. b_1_2 → c_1_4, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_2_7 → c_1_2·c_1_4, an element of degree 2
6. c_2_8 → c_1_2², an element of degree 2
7. c_2_9 → c_1_1·c_1_4 + c_1_1², an element of degree 2
8. c_2_10 → c_1_0·c_1_3 + c_1_0², 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 → 0, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_1_3 → c_1_4, an element of degree 1
5. b_2_7 → c_1_2·c_1_3, an element of degree 2
6. c_2_8 → c_1_2², an element of degree 2
7. c_2_9 → c_1_1·c_1_3 + c_1_1², an element of degree 2
8. c_2_10 → c_1_0·c_1_4 + c_1_0², an element of degree 2

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128