David J. Green

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Cohomology of group number 17 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 3.
• Its center has rank 2.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  ( − 1) · (t2  +  1)

  (t  −  1)3 · (t2  +  t  +  1)

• The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 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_0, a nilpotent element of degree 2
4. b_2_1, an element of degree 2
5. b_2_2, an element of degree 2
6. c_2_3, a Duflot regular 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. a_3_6, a nilpotent element of degree 3
10. a_4_6, a nilpotent element of degree 4
11. a_5_13, a nilpotent element of degree 5
12. a_6_13, a nilpotent element of degree 6
13. c_6_18, a Duflot regular element of degree 6

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Ring relations

There are 6 "obvious" relations:
   a_1_0², a_1_1², a_3_4², a_3_5², a_3_6², a_5_13²

Apart from that, there are 45 minimal relations of maximal degree 12:

1. a_1_0·a_1_1
2. a_2_0·a_1_1
3. a_2_0·a_1_0
4. b_2_1·a_1_0
5. b_2_2·a_1_1 + b_2_2·a_1_0
6. a_2_0²
7. a_2_0·b_2_1
8. a_2_0·b_2_2 + a_1_1·a_3_4
9.  − a_2_0·b_2_2 + a_1_0·a_3_4
10.  − b_2_1·b_2_2 + a_1_1·a_3_5
11. a_1_0·a_3_5
12. a_1_0·a_3_6
13. b_2_1·a_3_4
14. a_2_0·a_3_4
15. b_2_2·a_3_5 − b_2_2²·a_1_0
16. a_2_0·a_3_5
17. a_2_0·a_3_6
18. a_4_6·a_1_1
19. a_4_6·a_1_0
20. a_3_4·a_3_5 + b_2_2·a_1_0·a_3_4
21. a_3_4·a_3_6
22. b_2_1·a_4_6 − b_2_1·a_1_1·a_3_5
23. a_2_0·a_4_6
24.  − b_2_2·a_4_6 + a_1_1·a_5_13 − b_2_2·a_1_0·a_3_4 + b_2_1·a_1_1·a_3_5 − c_2_3·a_1_0·a_3_4
25. b_2_2·a_4_6 + a_1_0·a_5_13 + b_2_2·a_1_0·a_3_4 + c_2_3·a_1_0·a_3_4
26. a_4_6·a_3_6 − a_1_1·a_3_5·a_3_6
27. a_4_6·a_3_5
28. b_2_2²·a_3_6 + a_4_6·a_3_4
29. b_2_1·a_5_13 + b_2_1²·a_3_5 + a_1_1·a_3_5·a_3_6
30. b_2_2²·a_3_6 + a_2_0·a_5_13
31.  − b_2_2²·a_3_6 + a_6_13·a_1_1
32. b_2_2²·a_3_6 + a_6_13·a_1_0
33. a_4_6²
34. a_3_6·a_5_13 − b_2_1·a_3_5·a_3_6
35. a_3_5·a_5_13 − b_2_2·a_1_0·a_5_13
36. b_2_1·a_6_13 − b_2_1²·a_1_1·a_3_5
37. b_2_2·a_6_13 − a_3_4·a_5_13 − b_2_2·a_1_0·a_5_13 + b_2_2²·a_1_0·a_3_4
       − c_2_3·a_1_0·a_5_13 + b_2_2·c_2_3·a_1_0·a_3_4 − c_2_3²·a_1_0·a_3_4
38. a_2_0·a_6_13
39. a_4_6·a_5_13 + a_1_0·a_3_4·a_5_13 + a_2_0·c_2_3·a_5_13
40. a_6_13·a_3_6 − b_2_1·a_1_1·a_3_5·a_3_6
41. a_6_13·a_3_5 − a_1_0·a_3_4·a_5_13
42. a_6_13·a_3_4 + a_1_0·a_3_4·a_5_13 + a_2_0·c_2_3·a_5_13
43. a_4_6·a_6_13
44. a_6_13·a_5_13 + b_2_2·a_1_0·a_3_4·a_5_13 + c_2_3·a_1_0·a_3_4·a_5_13
       − a_2_0·c_2_3²·a_5_13
45. a_6_13²

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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_2_3, a Duflot regular element of degree 2
  2. c_6_18, a Duflot regular element of degree 6
  3. b_2_2² + b_2_1², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 9].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is 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_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. c_2_3 → c_2_1, 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. a_3_6 → 0, an element of degree 3
10. a_4_6 → 0, an element of degree 4
11. a_5_13 → 0, an element of degree 5
12. a_6_13 → 0, an element of degree 6
13. c_6_18 →  − c_2_2³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → a_1_2, an element of degree 1
3. a_2_0 → 0, an element of degree 2
4. b_2_1 → c_2_5, an element of degree 2
5. b_2_2 →  − a_1_1·a_1_2, an element of degree 2
6. c_2_3 → c_2_3, an element of degree 2
7. a_3_4 → 0, an element of degree 3
8. a_3_5 → c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
9. a_3_6 →  − c_2_5·a_1_0, an element of degree 3
10. a_4_6 →  − c_2_5·a_1_1·a_1_2, an element of degree 4
11. a_5_13 →  − c_2_5·a_1_0·a_1_1·a_1_2 − c_2_5²·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
12. a_6_13 →  − c_2_5²·a_1_1·a_1_2, an element of degree 6
13. c_6_18 →  − c_2_5²·a_1_1·a_1_2 − c_2_3²·a_1_1·a_1_2 + c_2_4·c_2_5² − c_2_4³, an element of degree 6

Restriction map to a maximal el. ab. subgp. 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_2_0 → 0, an element of degree 2
4. b_2_1 → 0, an element of degree 2
5. b_2_2 →  − c_2_5, an element of degree 2
6. c_2_3 → c_2_3, an element of degree 2
7. a_3_4 →  − c_2_5·a_1_2, an element of degree 3
8. a_3_5 → 0, an element of degree 3
9. a_3_6 → 0, an element of degree 3
10. a_4_6 → 0, an element of degree 4
11. a_5_13 → c_2_5²·a_1_1 − c_2_5²·a_1_0 − c_2_4·c_2_5·a_1_2 + c_2_3·c_2_5·a_1_2, an element of degree 5
12. a_6_13 →  − c_2_5²·a_1_1·a_1_2 + c_2_5²·a_1_0·a_1_2, an element of degree 6
13. c_6_18 →  − c_2_5³ + c_2_4·c_2_5² − c_2_4³ − c_2_3·c_2_5² − c_2_3²·c_2_5, an element of degree 6

About the group

Ring generators

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