David J. Green

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Cohomology of group number 15 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 a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t2  +  1)

  (t  +  1) · (t  −  1)3 · (t2  −  t  +  1) · (t2  +  t  +  1)

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

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Ring generators

The cohomology ring has 12 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. a_2_1, a nilpotent 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_4_4, a nilpotent element of degree 4
10. a_5_9, a nilpotent element of degree 5
11. a_6_7, a nilpotent element of degree 6
12. c_6_12, 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_4², a_3_5², a_5_9²

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

1. a_1_0·a_1_1
2. a_2_0·a_1_0
3. a_2_1·a_1_1 − a_2_0·a_1_1
4. a_2_1·a_1_0 − a_2_0·a_1_1
5. b_2_2·a_1_0 − a_2_0·a_1_1
6. a_2_0²
7. a_2_0·a_2_1
8. a_2_1²
9.  − a_2_0·b_2_2 + a_1_1·a_3_4
10. a_1_0·a_3_4
11.  − a_2_1·b_2_2 + a_1_1·a_3_5
12.  − a_2_0·b_2_2 + a_1_0·a_3_5
13. a_2_0·a_3_4 + a_2_0·c_2_3·a_1_1
14.  − b_2_2·a_3_4 + b_2_2²·a_1_1 + a_2_1·a_3_4 − b_2_2·c_2_3·a_1_1 + a_2_0·c_2_3·a_1_1
15.  − b_2_2·a_3_4 + b_2_2²·a_1_1 + a_2_0·a_3_5 − b_2_2·c_2_3·a_1_1 + a_2_0·c_2_3·a_1_1
16. b_2_2·a_3_4 − b_2_2²·a_1_1 + a_2_1·a_3_5 + b_2_2·c_2_3·a_1_1 + a_2_0·c_2_3·a_1_1
17.  − b_2_2·a_3_4 + b_2_2²·a_1_1 + a_4_4·a_1_1 − b_2_2·c_2_3·a_1_1
18. a_4_4·a_1_0 + a_2_0·c_2_3·a_1_1
19.  − a_3_4·a_3_5 + b_2_2·a_1_1·a_3_5 − c_2_3·a_1_1·a_3_5 + c_2_3·a_1_0·a_3_5
20. a_2_0·a_4_4
21. a_2_1·a_4_4
22.  − b_2_2·a_4_4 − a_3_4·a_3_5 + a_1_1·a_5_9 − c_2_3·a_1_1·a_3_5
23. a_3_4·a_3_5 + a_1_0·a_5_9 − b_2_2·a_1_1·a_3_5 + c_2_3·a_1_1·a_3_5
24. a_4_4·a_3_4 − a_2_0·c_2_3·a_3_5 − a_2_0·c_2_3²·a_1_1
25. a_2_0·a_5_9 + a_2_0·c_2_3·a_3_5
26. a_4_4·a_3_5 + a_2_1·a_5_9 − a_2_0·c_2_3·a_3_5 + a_2_0·c_2_3²·a_1_1
27. a_6_7·a_1_1 + a_4_4·a_3_5 + a_2_0·c_2_3²·a_1_1
28. a_6_7·a_1_0
29. a_4_4²
30. a_3_4·a_5_9 − b_2_2·a_1_1·a_5_9 + c_2_3·a_1_1·a_5_9 + c_2_3²·a_1_0·a_3_5
31. b_2_2·a_6_7 − a_3_5·a_5_9 + b_2_2·a_1_1·a_5_9 − b_2_2·c_2_3·a_1_1·a_3_5
       − c_2_3²·a_1_0·a_3_5
32. a_2_0·a_6_7
33. a_2_1·a_6_7
34. a_4_4·a_5_9 + a_1_1·a_3_5·a_5_9
35. a_6_7·a_3_5 + a_4_4·a_5_9 − a_2_0·c_6_12·a_1_1 − a_2_0·c_2_3³·a_1_1
36. a_6_7·a_3_4 + a_4_4·a_5_9 + a_2_1·c_2_3·a_5_9 − a_2_0·c_2_3²·a_3_5
37. a_4_4·a_6_7
38. a_6_7·a_5_9 − c_2_3·a_1_1·a_3_5·a_5_9 + a_2_0·c_2_3·c_6_12·a_1_1 + a_2_0·c_2_3⁴·a_1_1
39. a_6_7²

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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_12, a Duflot regular element of degree 6
  3. b_2_2, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

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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. a_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_4_4 → 0, an element of degree 4
10. a_5_9 → 0, an element of degree 5
11. a_6_7 → 0, an element of degree 6
12. c_6_12 → 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 → 0, an element of degree 1
3. a_2_0 → 0, an element of degree 2
4. a_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 → 0, an element of degree 3
8. a_3_5 → c_2_5·a_1_2, an element of degree 3
9. a_4_4 → 0, an element of degree 4
10. a_5_9 →  − c_2_5²·a_1_2 + c_2_5²·a_1_0 − c_2_3·c_2_5·a_1_2, an element of degree 5
11. a_6_7 → c_2_5²·a_1_0·a_1_2, an element of degree 6
12. c_6_12 → c_2_5²·a_1_0·a_1_2 − c_2_5³ − c_2_4·c_2_5² + c_2_4³, an element of degree 6

About the group

Ring generators

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Completion information

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Back to groups of order 243