David J. Green

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Cohomology of group number 1988 of order 128

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General information on the group

• The group has 4 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has 3 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) · (t4  −  t3  +  t2  +  t  +  1)

  (t  −  1)3 · (t2  +  1)2

• The a-invariants are -∞,-∞,-4,-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_3, a nilpotent element of degree 1
2. b_1_0, an element of degree 1
3. b_1_1, an element of degree 1
4. b_1_2, an element of degree 1
5. b_4_8, an element of degree 4
6. b_4_9, an element of degree 4
7. c_4_10, a Duflot regular element of degree 4
8. c_4_11, a Duflot regular element of degree 4

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Ring relations

There are 10 minimal relations of maximal degree 8:

1. b_1_0·b_1_2
2. b_1_1·b_1_2 + b_1_0·b_1_1 + a_1_3·b_1_0 + a_1_3²
3. b_1_0·b_1_1² + a_1_3·b_1_0·b_1_1 + a_1_3²·b_1_1 + a_1_3²·b_1_0
4. a_1_3·b_1_0·b_1_1 + a_1_3²·b_1_2 + a_1_3²·b_1_1
5. b_4_8·b_1_2
6. b_4_9·b_1_0 + b_4_8·b_1_0 + a_1_3·b_1_0⁴
7. b_4_9·b_1_1 + b_4_8·a_1_3
8. b_4_8·b_1_0⁴ + b_4_8² + c_4_11·b_1_0⁴ + c_4_10·b_1_0⁴
9. b_4_9² + b_4_8·b_1_0⁴ + c_4_11·b_1_2⁴ + c_4_11·b_1_0⁴ + c_4_10·b_1_0⁴
10. b_4_8·b_1_0⁴ + b_4_8·b_4_9 + b_4_8·a_1_3·b_1_0³ + c_4_11·b_1_0⁴ + c_4_10·b_1_0⁴

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_4_10, a Duflot regular element of degree 4
  2. c_4_11, a Duflot regular element of degree 4
  3. b_1_2² + b_1_1² + b_1_0², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

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Restriction maps

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

1. a_1_3 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_2 → 0, an element of degree 1
5. b_4_8 → 0, an element of degree 4
6. b_4_9 → 0, an element of degree 4
7. c_4_10 → c_1_0⁴, an element of degree 4
8. c_4_11 → c_1_1⁴ + c_1_0⁴, an element of degree 4

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

1. a_1_3 → 0, an element of degree 1
2. b_1_0 → c_1_2, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_2 → 0, an element of degree 1
5. b_4_8 → c_1_1·c_1_2³ + c_1_1²·c_1_2², an element of degree 4
6. b_4_9 → c_1_1·c_1_2³ + c_1_1²·c_1_2², an element of degree 4
7. c_4_10 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_11 → c_1_1·c_1_2³ + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

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

1. a_1_3 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_2 → c_1_2, an element of degree 1
5. b_4_8 → 0, an element of degree 4
6. b_4_9 → c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_2³ + c_1_0²·c_1_2², an element of degree 4
7. c_4_10 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_11 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

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

1. a_1_3 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → c_1_2, an element of degree 1
4. b_1_2 → 0, an element of degree 1
5. b_4_8 → 0, an element of degree 4
6. b_4_9 → 0, an element of degree 4
7. c_4_10 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_11 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128