David J. Green

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Cohomology of group number 62 of order 243

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General information on the group

• The group is also known as E27xV9, the Direct product E27 x C_3 x C_3.
• The group has 4 minimal generators and exponent 3.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 3.
• It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 4.
• The depth exceeds the Duflot bound, which is 3.
• The Poincaré series is

  (t2  +  1)2

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

• The a-invariants are -∞,-∞,-∞,-∞,-4. 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_1_2, a nilpotent element of degree 1
4. a_1_3, a nilpotent element of degree 1
5. b_2_5, an element of degree 2
6. b_2_6, an element of degree 2
7. b_2_7, an element of degree 2
8. b_2_8, an element of degree 2
9. c_2_9, a Duflot regular element of degree 2
10. c_2_10, a Duflot regular element of degree 2
11. a_3_22, a nilpotent element of degree 3
12. a_3_23, a nilpotent element of degree 3
13. c_6_108, 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_1_3², a_3_22², a_3_23²

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

1. a_1_0·a_1_1
2. b_2_6·a_1_0 − b_2_5·a_1_1
3. b_2_7·a_1_1 + b_2_6·a_1_1 − b_2_5·a_1_1
4. b_2_7·a_1_0 − b_2_6·a_1_1 + b_2_5·a_1_1
5. b_2_8·a_1_0 − b_2_5·a_1_1
6.  − b_2_6² + b_2_5·b_2_7 + b_2_5·b_2_6
7.  − b_2_6·b_2_7 − b_2_6² + b_2_5·b_2_8
8.  − b_2_7² + b_2_6·b_2_8 + b_2_6·b_2_7 − b_2_6²
9.  − b_2_7² + b_2_6·b_2_7 + a_1_1·a_3_22
10.  − b_2_6·b_2_7 − b_2_6² + b_2_5·b_2_6 + a_1_0·a_3_22
11.  − b_2_7·b_2_8 + b_2_7² + a_1_1·a_3_23
12.  − b_2_7² − b_2_6² + b_2_5·b_2_6 + a_1_0·a_3_23
13.  − b_2_8·a_3_22 + b_2_6·a_3_22
14.  − b_2_8·a_3_22 + b_2_5·a_3_23 − b_2_5·a_3_22 − b_2_5·b_2_6·a_1_1 + b_2_5²·a_1_1
15. b_2_7·a_3_23 + b_2_5·b_2_6·a_1_1 − b_2_5²·a_1_1
16. b_2_8·a_3_22 − b_2_7·a_3_22 + b_2_6·a_3_23 + b_2_5·b_2_6·a_1_1 − b_2_5²·a_1_1
17. a_3_22·a_3_23 + b_2_5·a_1_1·a_3_23 + b_2_5·a_1_0·a_3_23 − b_2_5·a_1_0·a_3_22

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_2_9, a Duflot regular element of degree 2
  2. c_2_10, a Duflot regular element of degree 2
  3. c_6_108, a Duflot regular element of degree 6
  4. b_2_8² − b_2_5·b_2_7 − b_2_5·b_2_6 + b_2_5², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 10].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
• We found that there exists some filter regular HSOP formed by the first 3 terms 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 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_1_3 → a_1_1, an element of degree 1
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_2_7 → 0, an element of degree 2
8. b_2_8 → 0, an element of degree 2
9. c_2_9 → c_2_3, an element of degree 2
10. c_2_10 → c_2_4, an element of degree 2
11. a_3_22 → 0, an element of degree 3
12. a_3_23 → 0, an element of degree 3
13. c_6_108 →  − c_2_5³, an element of degree 6

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

1. a_1_0 → a_1_3, 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_1_3 → a_1_1, an element of degree 1
5. b_2_5 → c_2_9, an element of degree 2
6. b_2_6 →  − a_1_2·a_1_3, an element of degree 2
7. b_2_7 → a_1_2·a_1_3, an element of degree 2
8. b_2_8 → 0, an element of degree 2
9. c_2_9 → c_2_6, an element of degree 2
10. c_2_10 → c_2_7, an element of degree 2
11. a_3_22 →  − c_2_9·a_1_2 + c_2_8·a_1_3, an element of degree 3
12. a_3_23 →  − c_2_9·a_1_2 + c_2_8·a_1_3, an element of degree 3
13. c_6_108 →  − c_2_9²·a_1_2·a_1_3 + c_2_8·c_2_9² − c_2_8³, 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_1_3 → a_1_1, an element of degree 1
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_2_7 → a_1_2·a_1_3, an element of degree 2
8. b_2_8 → c_2_9, an element of degree 2
9. c_2_9 → c_2_6, an element of degree 2
10. c_2_10 → c_2_7, an element of degree 2
11. a_3_22 → 0, an element of degree 3
12. a_3_23 →  − c_2_9·a_1_2 + c_2_8·a_1_3, an element of degree 3
13. c_6_108 → c_2_8·c_2_9² − c_2_8³, an element of degree 6

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

1. a_1_0 → a_1_3, 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_1_3 → a_1_1, an element of degree 1
5. b_2_5 → c_2_9, an element of degree 2
6. b_2_6 →  − a_1_2·a_1_3 + c_2_9, an element of degree 2
7. b_2_7 →  − a_1_2·a_1_3, an element of degree 2
8. b_2_8 → c_2_9, an element of degree 2
9. c_2_9 → c_2_6, an element of degree 2
10. c_2_10 → c_2_7, an element of degree 2
11. a_3_22 →  − c_2_9·a_1_2 + c_2_8·a_1_3, an element of degree 3
12. a_3_23 → c_2_9·a_1_2 − c_2_8·a_1_3, an element of degree 3
13. c_6_108 → c_2_8·c_2_9² − c_2_8³, an element of degree 6

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

1. a_1_0 →  − a_1_3, 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_1_3 → a_1_1, an element of degree 1
5. b_2_5 →  − c_2_9, an element of degree 2
6. b_2_6 → a_1_2·a_1_3 + c_2_9, an element of degree 2
7. b_2_7 → c_2_9, an element of degree 2
8. b_2_8 → c_2_9, an element of degree 2
9. c_2_9 → c_2_6, an element of degree 2
10. c_2_10 → c_2_7, an element of degree 2
11. a_3_22 → c_2_9·a_1_3 + c_2_9·a_1_2 − c_2_8·a_1_3, an element of degree 3
12. a_3_23 →  − c_2_9·a_1_3, an element of degree 3
13. c_6_108 → c_2_9²·a_1_2·a_1_3 + c_2_9³ + c_2_8·c_2_9² − c_2_8³, 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