David J. Green

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Cohomology of group number 6 of order 81

About the group

Ring generators

Ring relations

Completion information

Restriction maps

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General information on the group

• The group is also known as Mod81, the Modular group of order 81.
• The group has 2 minimal generators and exponent 27.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  1

  (t  −  1)2 · (t2  −  t  +  1) · (t2  +  t  +  1)

• The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

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Ring generators

The cohomology ring has 6 minimal generators of maximal degree 6:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_2_1, an element of degree 2
4. a_3_1, a nilpotent element of degree 3
5. a_5_1, a nilpotent element of degree 5
6. c_6_2, a Duflot regular element of degree 6

About the group

Ring generators

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Completion information

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Ring relations

There are 4 "obvious" relations:
   a_1_0², a_1_1², a_3_1², a_5_1²

Apart from that, there are 5 minimal relations of maximal degree 8:

1. b_2_1·a_1_0
2. a_1_0·a_3_1
3. b_2_1·a_3_1
4. a_1_0·a_5_1
5. a_3_1·a_5_1

About the group

Ring generators

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_6_2, a Duflot regular element of degree 6
  2. b_2_1², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, 4, 8].
• The filter degree type of any filter regular HSOP is [-1, -2, -2].
• We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 1 elements of degree 2.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 81

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_2_1 → 0, an element of degree 2
4. a_3_1 → 0, an element of degree 3
5. a_5_1 → 0, an element of degree 5
6. c_6_2 →  − c_2_0³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. b_2_1 → c_2_2, an element of degree 2
4. a_3_1 → 0, an element of degree 3
5. a_5_1 → 0, an element of degree 5
6. c_6_2 → c_2_1·c_2_2² − c_2_1³, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 81