David J. Green

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Cohomology of group number 28 of order 243

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General information on the group

• 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 3 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

  1

  (t  −  1)2

• 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 11 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. a_2_1, a nilpotent element of degree 2
4. b_2_0, an element of degree 2
5. b_2_2, an element of degree 2
6. a_3_2, a nilpotent element of degree 3
7. a_3_3, a nilpotent element of degree 3
8. b_4_4, an element of degree 4
9. a_5_5, a nilpotent element of degree 5
10. b_6_5, an element of degree 6
11. c_6_6, a Duflot regular element of degree 6

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Ring relations

There are 5 "obvious" relations:
   a_1_0², a_1_1², a_3_2², a_3_3², a_5_5²

Apart from that, there are 35 minimal relations of maximal degree 12:

1. a_1_0·a_1_1
2. a_2_1·a_1_1
3. a_2_1·a_1_0
4. b_2_2·a_1_1 + b_2_0·a_1_1
5. b_2_2·a_1_0 − b_2_0·a_1_1
6. a_2_1²
7. a_2_1·b_2_0
8. b_2_2² + b_2_0·b_2_2
9. a_2_1·b_2_2
10. a_1_1·a_3_2
11. a_1_0·a_3_2
12. b_2_0·a_3_2
13. b_2_2·a_3_2
14. a_2_1·a_3_2
15. a_2_1·a_3_3
16. b_4_4·a_1_0 + b_2_0²·a_1_1
17. b_2_0·b_4_4 + b_2_0²·b_2_2 + b_2_0·a_1_1·a_3_3 + b_2_0·a_1_0·a_3_3
18. b_2_2·b_4_4 − b_2_0²·b_2_2 − a_3_2·a_3_3
19. a_2_1·b_4_4 + a_3_2·a_3_3
20.  − a_3_2·a_3_3 + a_1_1·a_5_5 + b_2_0·a_1_1·a_3_3
21. a_1_0·a_5_5 + b_2_0·a_1_1·a_3_3 − b_2_0·a_1_0·a_3_3
22. b_2_0·a_5_5 + b_2_0·b_2_2·a_3_3 − b_2_0²·a_3_3
23. b_2_2·a_5_5 + b_2_0·b_2_2·a_3_3
24. a_2_1·a_5_5
25. b_6_5·a_1_1 − b_4_4·a_3_2
26. b_6_5·a_1_0
27. a_3_3·a_5_5 + b_4_4·a_1_1·a_3_3 − b_2_0²·a_1_1·a_3_3
28. a_3_2·a_5_5 − b_4_4·a_1_1·a_3_3 + b_2_0²·a_1_1·a_3_3
29. b_2_0·b_6_5 + b_2_0²·a_1_1·a_3_3
30. b_2_2·b_6_5 − b_4_4·a_1_1·a_3_3
31. a_2_1·b_6_5 + b_4_4·a_1_1·a_3_3 − b_2_0²·a_1_1·a_3_3
32. b_6_5·a_3_3 − b_4_4·a_5_5 + b_4_4²·a_1_1 + b_2_0²·b_2_2·a_3_3 − b_2_0⁴·a_1_1
33. b_6_5·a_3_2 − b_4_4²·a_1_1 + b_2_0⁴·a_1_1
34. b_6_5·a_5_5 − b_4_4²·a_3_3 − b_4_4²·a_3_2 − b_2_0³·b_2_2·a_3_3
35. b_6_5² − b_4_4³ − b_2_0⁵·b_2_2 + b_4_4²·a_1_1·a_3_3 − b_2_0⁴·a_1_1·a_3_3

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 12.
• 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_6, a Duflot regular element of degree 6
  2. b_4_4 + b_2_0·b_2_2 + 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].

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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. a_2_1 → 0, an element of degree 2
4. b_2_0 → 0, an element of degree 2
5. b_2_2 → 0, an element of degree 2
6. a_3_2 → 0, an element of degree 3
7. a_3_3 → 0, an element of degree 3
8. b_4_4 → 0, an element of degree 4
9. a_5_5 → 0, an element of degree 5
10. b_6_5 → 0, an element of degree 6
11. c_6_6 →  − 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. a_2_1 → 0, an element of degree 2
4. b_2_0 → c_2_2, an element of degree 2
5. b_2_2 → 0, an element of degree 2
6. a_3_2 → 0, an element of degree 3
7. a_3_3 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. b_4_4 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
9. a_5_5 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
10. b_6_5 → 0, an element of degree 6
11. c_6_6 → 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 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. b_2_0 → 0, an element of degree 2
5. b_2_2 → 0, an element of degree 2
6. a_3_2 → 0, an element of degree 3
7. a_3_3 → 0, an element of degree 3
8. b_4_4 → c_2_2², an element of degree 4
9. a_5_5 → 0, an element of degree 5
10. b_6_5 →  − c_2_2³, an element of degree 6
11. c_6_6 → 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. a_2_1 → 0, an element of degree 2
4. b_2_0 →  − c_2_2, an element of degree 2
5. b_2_2 → c_2_2, an element of degree 2
6. a_3_2 → 0, an element of degree 3
7. a_3_3 → c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. b_4_4 → c_2_2², an element of degree 4
9. a_5_5 → c_2_2²·a_1_1 − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
10. b_6_5 → c_2_2²·a_1_0·a_1_1, an element of degree 6
11. c_6_6 →  − c_2_2³ + c_2_1·c_2_2² − c_2_1³, an element of degree 6

About the group

Ring generators

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Completion information

Restriction maps

Back to groups of order 243