David J. Green

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Cohomology of group number 2251 of order 128

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General information on the group

• The group has 5 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• It has 6 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 exceeds the Duflot bound, which is 2.
• The Poincaré series is

  t4  +  t2  +  t  +  1

  (t  −  1)4 · (t2  +  1)2

• 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 8 minimal generators of maximal degree 4:

1. b_1_0, an 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_1_4, an element of degree 1
6. b_3_23, an element of degree 3
7. c_4_37, a Duflot regular element of degree 4
8. c_4_38, a Duflot regular element of degree 4

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Ring relations

There are 6 minimal relations of maximal degree 6:

1. b_1_4² + b_1_0·b_1_2
2. b_1_1·b_1_3 + b_1_0·b_1_4 + b_1_0·b_1_2 + b_1_0·b_1_1
3. b_1_0·b_1_2² + b_1_0²·b_1_2
4. b_1_0·b_1_3·b_1_4 + b_1_0·b_1_2·b_1_3 + b_1_0·b_1_1·b_1_4 + b_1_0·b_1_1·b_1_2
5. b_1_0·b_1_4·b_3_23 + b_1_0·b_1_2·b_3_23 + b_1_0³·b_1_2·b_1_4 + b_1_0⁴·b_1_2
6. b_3_23² + b_1_2·b_1_3²·b_3_23 + b_1_2²·b_1_4·b_3_23 + b_1_2³·b_3_23
      + b_1_1²·b_1_2·b_3_23 + b_1_0⁴·b_1_2·b_1_4 + c_4_38·b_1_3² + c_4_38·b_1_2²
      + c_4_38·b_1_1² + c_4_38·b_1_0·b_1_2 + c_4_37·b_1_2² + c_4_37·b_1_0·b_1_2

About the group

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_37, a Duflot regular element of degree 4
  2. c_4_38, a Duflot regular element of degree 4
  3. b_1_3·b_1_4 + b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_1·b_1_4 + b_1_1·b_1_2 + b_1_1²
        + b_1_0·b_1_3 + b_1_0·b_1_2 + b_1_0·b_1_1 + b_1_0², an element of degree 2
  4. b_1_3²·b_1_4 + b_1_2·b_1_3·b_1_4 + b_1_2·b_1_3² + b_1_2²·b_1_4 + b_1_2²·b_1_3
        + b_1_1·b_1_2·b_1_4 + b_1_1·b_1_2² + b_1_1²·b_1_4 + b_1_1²·b_1_2 + b_1_0·b_1_3²
        + b_1_0·b_1_2·b_1_4 + b_1_0·b_1_1² + b_1_0²·b_1_4 + b_1_0²·b_1_3 + b_1_0²·b_1_2
        + b_1_0²·b_1_1, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 9].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
• We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

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Restriction maps

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

1. b_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_1_4 → 0, an element of degree 1
6. b_3_23 → 0, an element of degree 3
7. c_4_37 → c_1_1⁴ + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_1⁴, an element of degree 4

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

1. b_1_0 → c_1_2, 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 → c_1_3, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. b_3_23 → c_1_1·c_1_2·c_1_3 + c_1_1²·c_1_3, an element of degree 3
7. c_4_37 → c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴
      + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. b_3_23 → c_1_1·c_1_2·c_1_3 + c_1_1²·c_1_2 + c_1_0·c_1_2·c_1_3 + c_1_0²·c_1_3, an element of degree 3
7. c_4_37 → c_1_1²·c_1_3² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3 + c_1_1⁴
      + c_1_0·c_1_2·c_1_3² + c_1_0²·c_1_3², an element of degree 4

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

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. b_3_23 → c_1_1·c_1_3² + c_1_1·c_1_2·c_1_3 + c_1_1²·c_1_3 + c_1_1²·c_1_2, an element of degree 3
7. c_4_37 → c_1_2·c_1_3³ + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_2·c_1_3 + c_1_1⁴
      + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4

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

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_1_4 → c_1_3, an element of degree 1
6. b_3_23 → c_1_3³ + c_1_1·c_1_3² + c_1_1·c_1_2·c_1_3 + c_1_1²·c_1_3 + c_1_1²·c_1_2, an element of degree 3
7. c_4_37 → c_1_3⁴ + c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1²·c_1_3² + c_1_1⁴
      + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_3⁴ + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_2·c_1_3 + c_1_1⁴, an element of degree 4

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

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. b_3_23 → c_1_1·c_1_2·c_1_3 + c_1_1²·c_1_3 + c_1_0·c_1_2·c_1_3 + c_1_0²·c_1_2, an element of degree 3
7. c_4_37 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴
      + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_2², an element of degree 4

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

1. b_1_0 → c_1_3, 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_2, an element of degree 1
5. b_1_4 → c_1_3, an element of degree 1
6. b_3_23 → c_1_3³ + c_1_1·c_1_2·c_1_3 + c_1_1²·c_1_2, an element of degree 3
7. c_4_37 → c_1_3⁴ + c_1_1²·c_1_3² + c_1_1⁴ + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3
      + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_38 → c_1_3⁴ + c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3 + 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