David J. Green

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Cohomology of group number 3 of order 27

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Ring generators

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General information on the group

• The group is also known as E27, the Extraspecial 3-group of order 27 and exponent 3.
• The group has 2 minimal generators and exponent 3.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  (t2  +  1)2

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

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

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Ring generators

The cohomology ring has 9 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_0, an element of degree 2
4. b_2_1, an element of degree 2
5. b_2_2, an element of degree 2
6. b_2_3, an element of degree 2
7. a_3_4, a nilpotent element of degree 3
8. a_3_5, a nilpotent element of degree 3
9. c_6_8, a Duflot regular element of degree 6

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Ring relations

There are 4 "obvious" relations:
   a_1_0², a_1_1², a_3_4², a_3_5²

Apart from that, there are 17 minimal relations of maximal degree 6:

1. a_1_0·a_1_1
2. b_2_1·a_1_0 − b_2_0·a_1_1
3. b_2_2·a_1_1 + b_2_1·a_1_1 − b_2_0·a_1_1
4. b_2_2·a_1_0 − b_2_1·a_1_1 + b_2_0·a_1_1
5. b_2_3·a_1_0 − b_2_0·a_1_1
6.  − b_2_1² + b_2_0·b_2_2 + b_2_0·b_2_1
7.  − b_2_2² + b_2_1·b_2_3 + b_2_1·b_2_2 − b_2_1²
8.  − b_2_1·b_2_2 − b_2_1² + b_2_0·b_2_3
9.  − b_2_2² + b_2_1·b_2_2 + a_1_1·a_3_4
10.  − b_2_1·b_2_2 − b_2_1² + b_2_0·b_2_1 + a_1_0·a_3_4
11.  − b_2_2·b_2_3 + b_2_2² + a_1_1·a_3_5
12.  − b_2_2² − b_2_1² + b_2_0·b_2_1 + a_1_0·a_3_5
13. b_2_3·a_3_4 − b_2_1·a_3_4
14.  − b_2_2·a_3_4 + b_2_1·a_3_5 + b_2_1·a_3_4 + b_2_0·b_2_1·a_1_1 − b_2_0²·a_1_1
15.  − b_2_1·a_3_4 + b_2_0·a_3_5 − b_2_0·a_3_4 − b_2_0·b_2_1·a_1_1 + b_2_0²·a_1_1
16. b_2_2·a_3_5 + b_2_0·b_2_1·a_1_1 − b_2_0²·a_1_1
17. a_3_4·a_3_5 + b_2_0·a_1_1·a_3_5 + b_2_0·a_1_0·a_3_5 − b_2_0·a_1_0·a_3_4

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 6.
• The following is a filter regular homogeneous system of parameters:
  1. c_6_8, a Duflot regular element of degree 6
  2. b_2_3² + b_2_2² − b_2_1·b_2_2 − b_2_1² + b_2_0², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, 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.

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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_0 → 0, an element of degree 2
4. b_2_1 → 0, an element of degree 2
5. b_2_2 → 0, an element of degree 2
6. b_2_3 → 0, an element of degree 2
7. a_3_4 → 0, an element of degree 3
8. a_3_5 → 0, an element of degree 3
9. c_6_8 →  − c_2_0³, an element of degree 6

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

1. a_1_0 → a_1_1, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_2_0 → c_2_2, an element of degree 2
4. b_2_1 →  − a_1_0·a_1_1, an element of degree 2
5. b_2_2 → a_1_0·a_1_1, an element of degree 2
6. b_2_3 → 0, an element of degree 2
7. a_3_4 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. a_3_5 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
9. c_6_8 →  − c_2_2²·a_1_0·a_1_1 + c_2_1·c_2_2² − c_2_1³, 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_0 → 0, an element of degree 2
4. b_2_1 → 0, an element of degree 2
5. b_2_2 → a_1_0·a_1_1, an element of degree 2
6. b_2_3 → c_2_2, an element of degree 2
7. a_3_4 → 0, an element of degree 3
8. a_3_5 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
9. c_6_8 → c_2_1·c_2_2² − c_2_1³, an element of degree 6

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

1. a_1_0 → a_1_1, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. b_2_0 → c_2_2, an element of degree 2
4. b_2_1 →  − a_1_0·a_1_1 + c_2_2, an element of degree 2
5. b_2_2 →  − a_1_0·a_1_1, an element of degree 2
6. b_2_3 → c_2_2, an element of degree 2
7. a_3_4 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. a_3_5 → c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
9. c_6_8 → c_2_1·c_2_2² − c_2_1³, an element of degree 6

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

1. a_1_0 →  − a_1_1, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. b_2_0 →  − c_2_2, an element of degree 2
4. b_2_1 → a_1_0·a_1_1 + c_2_2, an element of degree 2
5. b_2_2 → c_2_2, an element of degree 2
6. b_2_3 → c_2_2, an element of degree 2
7. a_3_4 → c_2_2·a_1_1 + c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
8. a_3_5 →  − c_2_2·a_1_1, an element of degree 3
9. c_6_8 → c_2_2²·a_1_0·a_1_1 + c_2_2³ + 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