David J. Green

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Cohomology of group number 63 of order 243

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General information on the group

• The group is also known as M27xV9, the Direct product M27 x C_3 x C_3.
• The group has 4 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

  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 10 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_6, an element of degree 2
6. c_2_7, a Duflot regular element of degree 2
7. c_2_8, a Duflot regular element of degree 2
8. a_3_15, a nilpotent element of degree 3
9. a_5_35, a nilpotent element of degree 5
10. c_6_49, 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_15², a_5_35²

Apart from that, there are 5 minimal relations of maximal degree 8:

1. b_2_6·a_1_0
2. a_1_0·a_3_15
3. b_2_6·a_3_15
4. a_1_0·a_5_35
5. a_3_15·a_5_35

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_7, a Duflot regular element of degree 2
  2. c_2_8, a Duflot regular element of degree 2
  3. c_6_49, a Duflot regular element of degree 6
  4. b_2_6, 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_1_3 → a_1_1, an element of degree 1
5. b_2_6 → 0, an element of degree 2
6. c_2_7 → c_2_3, an element of degree 2
7. c_2_8 → c_2_4, an element of degree 2
8. a_3_15 → 0, an element of degree 3
9. a_5_35 → 0, an element of degree 5
10. c_6_49 →  − c_2_5³, 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_6 → c_2_9, an element of degree 2
6. c_2_7 → c_2_6, an element of degree 2
7. c_2_8 → c_2_7, an element of degree 2
8. a_3_15 → 0, an element of degree 3
9. a_5_35 → c_2_9²·a_1_2 − c_2_8·c_2_9·a_1_3, an element of degree 5
10. c_6_49 → c_2_9²·a_1_2·a_1_3 + c_2_8·c_2_9² − c_2_8³, an element of degree 6

About the group

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