David J. Green

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Cohomology of group number 1033 of order 128

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Ring generators

Ring relations

Completion information

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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 4.
• Its center has rank 3.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t3  +  t  +  1

  (t  +  1) · (t  −  1)4 · (t2  +  1)

• The a-invariants are -∞,-∞,-∞,-5,-4. 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 4:

1. a_1_2, a nilpotent element of degree 1
2. a_1_0, a nilpotent element of degree 1
3. b_1_1, an element of degree 1
4. b_1_3, an element of degree 1
5. c_2_7, a Duflot regular element of degree 2
6. c_2_8, a Duflot regular element of degree 2
7. b_3_15, an element of degree 3
8. b_3_16, an element of degree 3
9. c_4_28, a Duflot regular element of degree 4

About the group

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Ring relations

There are 10 minimal relations of maximal degree 6:

1. a_1_0²
2. b_1_1·b_1_3 + a_1_0·b_1_1 + a_1_2²
3. a_1_2·b_1_3 + a_1_2·a_1_0
4. a_1_2²·b_1_1
5. a_1_2·b_3_15
6. b_1_3·b_3_16 + a_1_0·b_3_16
7. b_1_1·b_3_15 + a_1_2·b_3_16
8. b_3_15·b_3_16 + c_2_7·a_1_2·b_1_1³
9. b_3_15² + b_1_3³·b_3_15 + a_1_0·b_1_3²·b_3_15 + c_4_28·b_1_3²
10. b_3_16² + a_1_2·b_1_1²·b_3_16 + c_2_7·b_1_1⁴ + c_4_28·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 6.
• 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_2_7, a Duflot regular element of degree 2
  2. c_2_8, a Duflot regular element of degree 2
  3. c_4_28, a Duflot regular element of degree 4
  4. b_1_3² + b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3, 6].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

1. a_1_2 → 0, an element of degree 1
2. a_1_0 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. c_2_7 → c_1_1², an element of degree 2
6. c_2_8 → c_1_2², an element of degree 2
7. b_3_15 → 0, an element of degree 3
8. b_3_16 → 0, an element of degree 3
9. c_4_28 → c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. a_1_2 → 0, an element of degree 1
2. a_1_0 → 0, an element of degree 1
3. b_1_1 → c_1_3, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. c_2_7 → c_1_1², an element of degree 2
6. c_2_8 → c_1_2², an element of degree 2
7. b_3_15 → 0, an element of degree 3
8. b_3_16 → c_1_1·c_1_3², an element of degree 3
9. c_4_28 → c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. a_1_2 → 0, an element of degree 1
2. a_1_0 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. c_2_7 → c_1_1·c_1_3 + c_1_1², an element of degree 2
6. c_2_8 → c_1_3² + c_1_2², an element of degree 2
7. b_3_15 → c_1_0·c_1_3² + c_1_0²·c_1_3, an element of degree 3
8. b_3_16 → 0, an element of degree 3
9. c_4_28 → c_1_0·c_1_3³ + 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