David J. Green

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Cohomology of group number 372 of order 128

About the group

Ring generators

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General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• 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 coincides with the Duflot bound.
• The Poincaré series is

  − 1

  (t  −  1)3 · (t2  +  1)

• The a-invariants are -∞,-∞,-3,-3. 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 4:

1. a_1_0, a nilpotent element of degree 1
2. a_1_2, a nilpotent element of degree 1
3. b_1_1, an element of degree 1
4. b_2_4, an element of degree 2
5. a_3_3, a nilpotent element of degree 3
6. b_3_6, an element of degree 3
7. c_4_8, a Duflot regular element of degree 4
8. c_4_9, a Duflot regular element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

1. a_1_0·a_1_2
2. a_1_0·b_1_1
3. a_1_2³ + a_1_0³
4. a_1_2²·b_1_1
5. b_2_4·a_1_0
6. b_2_4·b_1_1² + b_2_4² + b_2_4·a_1_2·b_1_1
7. b_1_1·a_3_3
8. a_1_2·a_3_3 + b_2_4·a_1_2²
9. a_1_0·b_3_6 + b_2_4·a_1_2²
10. b_2_4·a_3_3
11. a_1_2²·b_3_6 + a_1_0²·a_3_3
12. a_3_3·b_3_6
13. a_3_3² + c_4_8·a_1_0²
14. b_3_6² + a_1_2·b_1_1²·b_3_6 + b_2_4²·a_1_2·b_1_1 + c_4_9·b_1_1² + c_4_8·a_1_2²

About the group

Ring generators

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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_8, a Duflot regular element of degree 4
  2. c_4_9, a Duflot regular element of degree 4
  3. b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. a_1_2 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. a_3_3 → 0, an element of degree 3
6. b_3_6 → 0, an element of degree 3
7. c_4_8 → c_1_0⁴, an element of degree 4
8. c_4_9 → c_1_1⁴, an element of degree 4

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

1. a_1_0 → 0, an element of degree 1
2. a_1_2 → 0, an element of degree 1
3. b_1_1 → c_1_2, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. a_3_3 → 0, an element of degree 3
6. b_3_6 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
7. c_4_8 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_9 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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

1. a_1_0 → 0, an element of degree 1
2. a_1_2 → 0, an element of degree 1
3. b_1_1 → c_1_2, an element of degree 1
4. b_2_4 → c_1_2², an element of degree 2
5. a_3_3 → 0, an element of degree 3
6. b_3_6 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
7. c_4_8 → c_1_2⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_9 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128