David J. Green

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Cohomology of group number 2231 of order 128

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Ring generators

Ring relations

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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 3.
• Its center has rank 2.
• It has 7 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 3.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  ( − 1) · (t2  +  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)

  (t  −  1)3 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.

About the group

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Ring generators

The cohomology ring has 7 minimal generators of maximal degree 8:

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. c_2_13, a Duflot regular element of degree 2
7. c_8_165, a Duflot regular element of degree 8

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Ring relations

There are 4 minimal relations of maximal degree 5:

1. b_1_2² + b_1_0·b_1_2
2. b_1_4² + b_1_1·b_1_3 + b_1_0·b_1_4 + b_1_0·b_1_1 + b_1_0²
3. b_1_1·b_1_3² + b_1_1²·b_1_3 + b_1_0·b_1_1·b_1_3 + b_1_0·b_1_1² + b_1_0³
4. b_1_0³·b_1_3² + b_1_0³·b_1_1·b_1_3 + b_1_0³·b_1_1² + b_1_0⁴·b_1_3 + b_1_0⁵

About the group

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 9.
• However, the last relation was already found in degree 5 and the last generator in degree 8.
• The following is a filter regular homogeneous system of parameters:
  1. c_2_13, a Duflot regular element of degree 2
  2. c_8_165, a Duflot regular element of degree 8
  3. b_1_3·b_1_4 + b_1_3² + b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 9].
• 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. 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. c_2_13 → c_1_1², an element of degree 2
7. c_8_165 → c_1_0⁸, an element of degree 8

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

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 → 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. c_2_13 → c_1_1², an element of degree 2
7. c_8_165 → c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

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

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 → c_1_2, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. c_2_13 → c_1_1², an element of degree 2
7. c_8_165 → c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

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

1. b_1_0 → c_1_2, 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 → 0, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. c_2_13 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. c_8_165 → c_1_2⁸ + c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

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

1. b_1_0 → c_1_2, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → c_1_2, 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. c_2_13 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
7. c_8_165 → c_1_2⁸ + c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

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

1. b_1_0 → c_1_2, 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 → 0, an element of degree 1
5. b_1_4 → c_1_2, an element of degree 1
6. c_2_13 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. c_8_165 → c_1_2⁸ + c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

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

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 → 0, an element of degree 1
4. b_1_3 → c_1_2, an element of degree 1
5. b_1_4 → c_1_2, an element of degree 1
6. c_2_13 → c_1_1², an element of degree 2
7. c_8_165 → c_1_2⁸ + c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

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

1. b_1_0 → c_1_2, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_1_4 → c_1_2, an element of degree 1
6. c_2_13 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
7. c_8_165 → c_1_2⁸ + c_1_0⁴·c_1_2⁴ + 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