David J. Green

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Cohomology of group number 2185 of order 128

About the group

Ring generators

Ring relations

Completion information

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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 3.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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

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

  (t  −  1)4 · (t2  +  1)2

• The a-invariants are -∞,-∞,-∞,-4,-4. 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 8 minimal generators of maximal degree 5:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_3, a nilpotent element of degree 1
4. b_1_2, an element of degree 1
5. c_1_4, a Duflot regular element of degree 1
6. c_4_31, a Duflot regular element of degree 4
7. c_4_32, a Duflot regular element of degree 4
8. b_5_45, an element of degree 5

About the group

Ring generators

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Ring relations

There are 7 minimal relations of maximal degree 10:

1. a_1_1·b_1_2 + a_1_3² + a_1_0·a_1_3 + a_1_0·a_1_1 + a_1_0²
2. a_1_0·b_1_2 + a_1_1²
3. a_1_1·a_1_3² + a_1_0·a_1_1·a_1_3 + a_1_0²·a_1_1
4. a_1_3²·b_1_2 + a_1_1²·a_1_3 + a_1_0·a_1_1² + a_1_0³
5. a_1_1·b_5_45 + c_4_32·a_1_1² + c_4_31·a_1_1·a_1_3
6. a_1_0·b_5_45 + c_4_32·a_1_0·a_1_1 + c_4_31·a_1_0·a_1_3
7. b_5_45² + c_4_31·b_1_2⁶ + c_4_32²·a_1_1² + c_4_31²·a_1_3²

About the group

Ring generators

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• 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_1_4, a Duflot regular element of degree 1
  2. c_4_31, a Duflot regular element of degree 4
  3. c_4_32, a Duflot regular element of degree 4
  4. b_1_2², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

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Restriction maps

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

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_3 → 0, an element of degree 1
4. b_1_2 → 0, an element of degree 1
5. c_1_4 → c_1_0, an element of degree 1
6. c_4_31 → c_1_2⁴, an element of degree 4
7. c_4_32 → c_1_1⁴, an element of degree 4
8. b_5_45 → 0, an element of degree 5

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

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_3 → 0, an element of degree 1
4. b_1_2 → c_1_3, an element of degree 1
5. c_1_4 → c_1_0, an element of degree 1
6. c_4_31 → c_1_2²·c_1_3² + c_1_2⁴, an element of degree 4
7. c_4_32 → c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4
8. b_5_45 → c_1_2·c_1_3⁴ + c_1_2²·c_1_3³, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128