David J. Green

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Cohomology of group number 697 of order 128

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General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• 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 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t2  −  t  +  1

  (t  −  1)4 · (t2  +  1)

• The a-invariants are -∞,-∞,-4,-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 4:

1. a_1_0, a nilpotent element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. b_2_3, an element of degree 2
5. b_2_4, an element of degree 2
6. c_2_5, a Duflot regular element of degree 2
7. b_3_9, an element of degree 3
8. b_3_10, an element of degree 3
9. b_4_15, an element of degree 4
10. c_4_18, a Duflot regular element of degree 4

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Ring relations

There are 19 minimal relations of maximal degree 8:

1. a_1_0²
2. a_1_0·b_1_1
3. b_1_2² + b_1_1·b_1_2 + a_1_0·b_1_2
4. b_2_3·a_1_0
5. b_2_4·a_1_0
6. b_2_3·b_1_1² + b_2_3²
7. a_1_0·b_3_9
8. a_1_0·b_3_10
9. b_1_1·b_3_10 + b_1_1⁴ + b_2_3·b_2_4
10. b_2_3·b_3_10 + b_2_3·b_2_4·b_1_1 + b_2_3²·b_1_1
11. b_4_15·a_1_0
12. b_1_1²·b_3_9 + b_1_1⁵ + b_4_15·b_1_1 + b_2_4·b_1_1³ + b_2_3·b_3_9 + b_2_3²·b_1_1
13. b_3_10² + b_1_1⁶ + b_2_3·b_2_4²
14. b_3_9² + b_1_1⁶ + b_4_15·b_1_1² + b_2_4·b_1_1·b_3_9 + b_2_4·b_1_1⁴ + b_2_4³
       + b_2_3·b_1_1·b_3_9 + b_2_3²·b_2_4 + c_4_18·b_1_1²
15. b_2_3·b_4_15 + b_2_3²·b_2_4
16. b_3_9·b_3_10 + b_3_9² + b_2_4·b_1_1⁴ + b_2_4·b_4_15 + b_2_4²·b_1_1² + b_2_4³
       + b_2_3³ + c_4_18·b_1_1²
17. b_4_15·b_3_10 + b_4_15·b_1_1³ + b_2_3·b_2_4²·b_1_1
18. b_4_15·b_3_9 + b_2_4²·b_3_10 + b_2_4²·b_1_1³ + b_2_4³·b_1_1 + b_2_3·b_2_4·b_3_9
       + c_4_18·b_1_1³ + b_2_3·c_4_18·b_1_1
19. b_4_15·b_1_1⁴ + b_4_15² + b_2_4·b_4_15·b_1_1² + b_2_4³·b_1_1² + b_2_3·b_2_4³
       + b_2_3³·b_2_4 + c_4_18·b_1_1⁴ + b_2_3²·c_4_18

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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_5, a Duflot regular element of degree 2
  2. c_4_18, a Duflot regular element of degree 4
  3. b_1_1² + b_2_4, an element of degree 2
  4. b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 2, 4, 6].
• 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 2

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. c_2_5 → c_1_0², an element of degree 2
7. b_3_9 → 0, an element of degree 3
8. b_3_10 → 0, an element of degree 3
9. b_4_15 → 0, an element of degree 4
10. c_4_18 → c_1_1⁴, an element of degree 4

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → c_1_2·c_1_3 + c_1_2², an element of degree 2
6. c_2_5 → c_1_0·c_1_3 + c_1_0², an element of degree 2
7. b_3_9 → c_1_2·c_1_3² + c_1_2³ + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
8. b_3_10 → c_1_3³, an element of degree 3
9. b_4_15 → c_1_3⁴ + c_1_2²·c_1_3² + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1²·c_1_3², an element of degree 4
10. c_4_18 → c_1_2·c_1_3³ + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3²
       + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → c_1_2·c_1_3 + c_1_2², an element of degree 2
6. c_2_5 → c_1_3² + c_1_0·c_1_3 + c_1_0², an element of degree 2
7. b_3_9 → c_1_2·c_1_3² + c_1_2³ + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
8. b_3_10 → c_1_3³, an element of degree 3
9. b_4_15 → c_1_3⁴ + c_1_2²·c_1_3² + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1²·c_1_3², an element of degree 4
10. c_4_18 → c_1_2·c_1_3³ + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3²
       + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_3 → c_1_3², an element of degree 2
5. b_2_4 → c_1_2·c_1_3 + c_1_2², an element of degree 2
6. c_2_5 → c_1_0·c_1_3 + c_1_0², an element of degree 2
7. b_3_9 → c_1_3³ + c_1_2²·c_1_3 + c_1_2³ + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
8. b_3_10 → c_1_3³ + c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
9. b_4_15 → c_1_2·c_1_3³ + c_1_2²·c_1_3², an element of degree 4
10. c_4_18 → c_1_3⁴ + c_1_2²·c_1_3² + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3²
       + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_2_3 → c_1_3², an element of degree 2
5. b_2_4 → c_1_2·c_1_3 + c_1_2², an element of degree 2
6. c_2_5 → c_1_3² + c_1_0·c_1_3 + c_1_0², an element of degree 2
7. b_3_9 → c_1_3³ + c_1_2²·c_1_3 + c_1_2³ + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
8. b_3_10 → c_1_3³ + c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
9. b_4_15 → c_1_2·c_1_3³ + c_1_2²·c_1_3², an element of degree 4
10. c_4_18 → c_1_3⁴ + c_1_2²·c_1_3² + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3²
       + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128