David J. Green

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Singular

Gap

Cohomology of group number 33 of order 243

About the group

Ring generators

Ring relations

Completion information

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General information on the group

• The group is also known as 81gp4xC3, the Direct product 81gp4 x C_3.
• The group has 3 minimal generators and exponent 9.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 3.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  − 1

  (t  −  1)3

• The a-invariants are -∞,-∞,-∞,-3. 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 2:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_2, a nilpotent element of degree 1
4. c_2_3, a Duflot regular element of degree 2
5. c_2_4, a Duflot regular element of degree 2
6. c_2_5, a Duflot regular element of degree 2

About the group

Ring generators

Ring relations

Completion information

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Ring relations

There are 3 "obvious" relations:
   a_1_0², a_1_1², a_1_2²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Completion information

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 3.
• However, the last relation was already found in degree 0 and the last generator in degree 2.
• The following is a filter regular homogeneous system of parameters:
  1. c_2_3, a Duflot regular element of degree 2
  2. c_2_4, a Duflot regular element of degree 2
  3. c_2_5, a Duflot regular element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

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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_2 → a_1_0, an element of degree 1
4. c_2_3 → c_2_3, an element of degree 2
5. c_2_4 → c_2_5, an element of degree 2
6. c_2_5 → c_2_4, an element of degree 2

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243