David J. Green

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Cohomology of group number 235 of order 64

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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 3.
• Its center has rank 2.
• It has 2 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  +  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 7 minimal generators of maximal degree 4:

1. a_1_1, a nilpotent element of degree 1
2. a_1_0, a nilpotent element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. b_3_10, an element of degree 3
6. c_4_13, a Duflot regular element of degree 4
7. c_4_14, a Duflot regular element of degree 4

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Ring relations

There are 6 minimal relations of maximal degree 6:

1. a_1_0² + a_1_1·a_1_0 + a_1_1²
2. b_1_2·b_1_3 + a_1_0·b_1_2 + a_1_1²
3. a_1_1³
4. a_1_1²·b_1_3 + a_1_1²·b_1_2 + a_1_1²·a_1_0
5. a_1_1²·b_3_10
6. b_3_10² + b_1_3³·b_3_10 + a_1_0·b_1_3²·b_3_10 + a_1_1·b_1_3²·b_3_10
      + a_1_1·b_1_2²·b_3_10 + a_1_1·a_1_0·b_1_3·b_3_10 + c_4_14·b_1_3² + c_4_14·b_1_2²
      + c_4_13·b_1_3² + c_4_13·b_1_2² + c_4_14·a_1_1·a_1_0 + c_4_14·a_1_1²
      + c_4_13·a_1_1·a_1_0

About the group

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 7.
• 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_13, a Duflot regular element of degree 4
  2. c_4_14, a Duflot regular element of degree 4
  3. b_1_3 + b_1_2, an element of degree 1
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

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

1. a_1_1 → 0, an element of degree 1
2. a_1_0 → 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_3_10 → 0, an element of degree 3
6. c_4_13 → c_1_1⁴, an element of degree 4
7. c_4_14 → 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_1 → 0, an element of degree 1
2. a_1_0 → 0, 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_3_10 → c_1_0²·c_1_2, an element of degree 3
6. c_4_13 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
7. c_4_14 → c_1_1²·c_1_2² + 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_1 → 0, an element of degree 1
2. a_1_0 → 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_3_10 → c_1_2³ + c_1_0²·c_1_2, an element of degree 3
6. c_4_13 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2², an element of degree 4
7. c_4_14 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0⁴, an element of degree 4

About the group

Ring generators

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Completion information

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Back to groups of order 64