David J. Green

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Cohomology of group number 962 of order 128

About the group

Ring generators

Ring relations

Completion information

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General information on the group

• The group has 3 minimal generators and exponent 16.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  (t2  +  t  +  1) · (t6  +  t3  +  1)

  (t  −  1)2 · (t2  +  1)2 · (t4  +  1)

• The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 8:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_1_2, an element of degree 1
4. a_3_3, a nilpotent element of degree 3
5. b_4_4, an element of degree 4
6. a_5_6, a nilpotent element of degree 5
7. b_5_5, an element of degree 5
8. a_7_7, a nilpotent element of degree 7
9. c_8_9, a Duflot regular element of degree 8

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Ring relations

There are 21 minimal relations of maximal degree 14:

1. a_1_1² + a_1_0·a_1_1 + a_1_0²
2. a_1_0·b_1_2
3. a_1_0³
4. b_1_2·a_3_3
5. a_1_0²·a_3_3
6. b_4_4·b_1_2
7. b_1_2·a_5_6
8. a_3_3² + a_1_0·a_5_6 + b_4_4·a_1_0²
9. a_1_0·b_5_5
10. a_1_0²·a_5_6
11. a_3_3·b_5_5
12. b_1_2·a_7_7 + a_1_1·b_1_2²·b_5_5
13. a_3_3·a_5_6 + a_1_0·a_7_7
14. b_4_4·b_5_5 + a_1_0²·a_7_7 + b_4_4·a_1_0·a_1_1·a_3_3
15. a_5_6² + a_3_3·a_7_7 + b_4_4·a_1_0·a_5_6
16. a_5_6·b_5_5 + a_1_0²·a_1_1·a_7_7
17. b_5_5² + c_8_9·b_1_2²
18. a_5_6·b_5_5 + a_5_6² + b_4_4·a_1_0·a_5_6 + c_8_9·a_1_0²
19. a_5_6·a_7_7 + b_4_4·a_1_0·a_7_7 + c_8_9·a_1_0·a_3_3
20. b_5_5·a_7_7 + c_8_9·a_1_1·b_1_2³
21. a_7_7² + c_8_9·a_1_0·a_5_6

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 14.
• 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_9, a Duflot regular element of degree 8
  2. b_1_2⁴ + b_4_4, an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, 10].
• The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group

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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. a_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. a_3_3 → 0, an element of degree 3
5. b_4_4 → 0, an element of degree 4
6. a_5_6 → 0, an element of degree 5
7. b_5_5 → 0, an element of degree 5
8. a_7_7 → 0, an element of degree 7
9. c_8_9 → c_1_0⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_1, an element of degree 1
4. a_3_3 → 0, an element of degree 3
5. b_4_4 → 0, an element of degree 4
6. a_5_6 → 0, an element of degree 5
7. b_5_5 → c_1_0²·c_1_1³ + c_1_0⁴·c_1_1, an element of degree 5
8. a_7_7 → 0, an element of degree 7
9. c_8_9 → c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. a_3_3 → 0, an element of degree 3
5. b_4_4 → c_1_1⁴, an element of degree 4
6. a_5_6 → 0, an element of degree 5
7. b_5_5 → 0, an element of degree 5
8. a_7_7 → 0, an element of degree 7
9. c_8_9 → 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

Back to groups of order 128