David J. Green

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Singular

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Cohomology of group number 4 of order 27

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 M27, the Extraspecial 3-group of order 27 and exponent 9.
• The group has 2 minimal generators and exponent 9.
• 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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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

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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 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
6. c_6_2 → c_2_2²·a_1_0·a_1_1 + 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 27