David J. Green

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Cohomology of group number 32 of order 243

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General information on the group

• The group is also known as 81gp3xC3, the Direct product 81gp3 x C_3.
• The group has 3 minimal generators and exponent 9.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 3.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t2  +  1

  (t  +  1) · (t  −  1)4 · (t2  −  t  +  1) · (t2  +  t  +  1)

• The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 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_1_2, a nilpotent element of degree 1
4. a_2_2, a nilpotent element of degree 2
5. a_2_3, a nilpotent element of degree 2
6. b_2_4, an element of degree 2
7. c_2_5, a Duflot regular element of degree 2
8. c_2_6, a Duflot regular element of degree 2
9. a_3_11, a nilpotent element of degree 3
10. a_3_12, a nilpotent element of degree 3
11. a_4_17, a nilpotent element of degree 4
12. a_5_30, a nilpotent element of degree 5
13. a_6_41, a nilpotent element of degree 6
14. c_6_42, 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_1_2², a_3_11², a_3_12², a_5_30²

Apart from that, there are 39 minimal relations of maximal degree 12:

1. a_1_0·a_1_1
2. a_2_2·a_1_0
3. a_2_3·a_1_1 + a_2_2·a_1_1
4. a_2_3·a_1_0 − a_2_2·a_1_1
5. b_2_4·a_1_0
6. a_2_2²
7. a_2_2·a_2_3
8. a_2_2·b_2_4 − a_2_3²
9. a_1_1·a_3_11 − a_2_3²
10. a_1_0·a_3_11
11.  − a_2_3·b_2_4 + a_1_1·a_3_12 − a_2_3²
12. a_1_0·a_3_12 − a_2_3²
13. b_2_4·a_3_11
14. a_2_2·a_3_11
15.  − a_2_3·a_3_11 + a_2_2·a_3_12
16. a_2_3·a_3_12 + a_2_3·a_3_11
17. a_4_17·a_1_1 − a_2_3·a_3_11
18. a_4_17·a_1_0
19. a_3_11·a_3_12
20. a_2_2·a_4_17
21. a_2_3·a_4_17
22.  − b_2_4·a_4_17 + a_1_1·a_5_30 + b_2_4·a_1_1·a_3_12 − a_2_3²·c_2_6
23. a_1_0·a_5_30
24. a_4_17·a_3_11
25. a_2_2·a_5_30
26. a_4_17·a_3_12 + a_2_3·a_5_30 − a_2_2·c_2_6·a_3_12
27. a_6_41·a_1_1 + a_4_17·a_3_12 − a_2_2·c_2_6·a_3_12
28. a_6_41·a_1_0
29. a_4_17²
30. a_3_11·a_5_30
31. b_2_4·a_6_41 − a_3_12·a_5_30 − b_2_4²·a_1_1·a_3_12 − c_2_6·a_1_1·a_5_30
       − b_2_4·c_2_6·a_1_1·a_3_12 + a_2_3²·c_2_6²
32. a_2_2·a_6_41
33. a_2_3·a_6_41
34.  − a_4_17·a_5_30 + a_1_1·a_3_12·a_5_30
35. a_6_41·a_3_12 + a_2_3·c_2_6·a_5_30 − a_2_2·c_6_42·a_1_1 − a_2_2·c_2_6²·a_3_12
36. a_6_41·a_3_11
37. a_4_17·a_6_41
38. a_6_41·a_5_30 − b_2_4·a_1_1·a_3_12·a_5_30 − c_2_6·a_1_1·a_3_12·a_5_30
39. a_6_41²

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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_5, a Duflot regular element of degree 2
  2. c_2_6, a Duflot regular element of degree 2
  3. c_6_42, a Duflot regular element of degree 6
  4. b_2_4, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. a_2_2 → 0, an element of degree 2
5. a_2_3 → 0, an element of degree 2
6. b_2_4 → 0, an element of degree 2
7. c_2_5 → c_2_3, an element of degree 2
8. c_2_6 → c_2_5, an element of degree 2
9. a_3_11 → 0, an element of degree 3
10. a_3_12 → 0, an element of degree 3
11. a_4_17 → 0, an element of degree 4
12. a_5_30 → 0, an element of degree 5
13. a_6_41 → 0, an element of degree 6
14. c_6_42 →  − c_2_4³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → a_1_3, an element of degree 1
3. a_1_2 → a_1_0, an element of degree 1
4. a_2_2 → 0, an element of degree 2
5. a_2_3 → a_1_1·a_1_3, an element of degree 2
6. b_2_4 → c_2_9, an element of degree 2
7. c_2_5 → c_2_6, an element of degree 2
8. c_2_6 → c_2_8, an element of degree 2
9. a_3_11 → 0, an element of degree 3
10. a_3_12 →  − c_2_9·a_1_1 + c_2_7·a_1_3, an element of degree 3
11. a_4_17 →  − c_2_9·a_1_2·a_1_3 − c_2_9·a_1_1·a_1_3, an element of degree 4
12. a_5_30 → c_2_9²·a_1_2 − c_2_9²·a_1_1 + c_2_7·c_2_9·a_1_3, an element of degree 5
13. a_6_41 → c_2_9²·a_1_1·a_1_3 − c_2_9²·a_1_1·a_1_2 − c_2_8·c_2_9·a_1_2·a_1_3
       − c_2_8·c_2_9·a_1_1·a_1_3 − c_2_7·c_2_9·a_1_2·a_1_3, an element of degree 6
14. c_6_42 → c_2_9²·a_1_1·a_1_3 − c_2_8·c_2_9·a_1_2·a_1_3 + c_2_8·c_2_9² − c_2_8²·c_2_9
       + c_2_7·c_2_9² − c_2_7³, an element of degree 6

About the group

Ring generators

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Back to groups of order 243