Mathematics:

Cohomology
→Theory
→Implementation

Jena:

Faculty

David Green

External links:

Singular

Gap

Mod-2-Cohomology of AlternatingGroup(9), a group of order 181440

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• AlternatingGroup(9) is a group of order 181440.
• The group order factors as 2⁶ · 3⁴ · 5 · 7.
• The group is defined by Group([(1,2,3,4,5,6,7,8,9),(7,8,9)]).
• It is non-abelian.
• It has 2-Rank 4.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 3, 3 and 4, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  1  −  t  +  2·t2  −  t3  +  2·t4  +  2·t6  +  t7  +  t8  −  t9  +  t11  −  t12  +  t13  −  t14  +  t15

  (1  +  t) · ( − 1  +  t)4 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2)2 · (1  +  t  +  t2  +  t3  +  t4  +  t5  +  t6)

• The a-invariants are -∞,-∞,-∞,-6,-4. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 9 minimal generators of maximal degree 7:

1. b_2_0, an element of degree 2
2. b_3_1, an element of degree 3
3. b_3_0, an element of degree 3
4. c_4_1, a Duflot element of degree 4
5. b_5_2, an element of degree 5
6. b_6_2, an element of degree 6
7. b_6_1, an element of degree 6
8. b_7_6, an element of degree 7
9. b_7_5, an element of degree 7

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 17 minimal relations of maximal degree 14:

1. b_3_0·b_3_1
2. b_3_0·b_5_2
3. b_2_0·b_7_5
4. b_2_0·b_7_6
5. b_6_1·b_3_0
6. b_6_2·b_3_0
7. b_3_0·b_7_5
8. b_3_0·b_7_6
9. b_3_1·b_7_5
10. b_3_1·b_7_6
11. b_5_2² + b_2_0·b_3_1·b_5_2 + b_2_0²·b_6_2 + c_4_1·b_3_1²
12. b_6_1·b_6_2 + b_6_1² + b_2_0²·b_3_1·b_5_2 + b_2_0³·b_6_2 + c_4_1·b_3_1·b_5_2
       + b_2_0·c_4_1·b_3_1² + b_2_0·c_4_1·b_6_2
13. b_5_2·b_7_5
14. b_5_2·b_7_6
15. b_6_1·b_7_6 + b_6_1·b_7_5
16. b_6_2·b_7_5 + b_6_1·b_7_5
17. b_7_5·b_7_6 + b_7_5²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 20 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 14 and the last generator in degree 7.
• The following is a filter regular homogeneous system of parameters:
  1. b_3_1·b_5_2 + b_2_0⁴ + c_4_1², an element of degree 8
  2. b_3_1⁴ + b_3_0⁴ + b_6_2·b_3_1² + b_6_2² + b_2_0³·b_6_2 + c_4_1·b_3_1·b_5_2
        + b_2_0·c_4_1·b_3_1² + b_2_0²·c_4_1², an element of degree 12
  3. b_7_6² + b_3_1³·b_5_2 + b_6_2·b_3_1·b_5_2 + b_2_0·b_6_2·b_3_1²
        + b_2_0³·b_3_1·b_5_2 + b_2_0⁴·b_6_2 + b_2_0²·c_4_1·b_3_1² + b_2_0²·c_4_1·b_6_2
        + c_4_1²·b_3_0², an element of degree 14
  4. b_6_2, an element of degree 6
• A Duflot regular sequence is given by c_4_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 28, 36].
• Modifying the above filter regular HSOP, we obtained the following parameters:
1. b_2_0² + c_4_1, an element of degree 4
2. b_3_1² + b_3_0² + b_2_0·c_4_1, an element of degree 6
3. b_7_6 + c_4_1·b_3_1 + c_4_1·b_3_0, an element of degree 7
4. b_6_2, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(192,1493); GF(2))

1. b_2_0 → b_1_0² + b_2_2 + b_2_1
2. b_3_1 → b_3_0 + b_2_2·b_1_0 + b_2_1·b_1_0
3. b_3_0 → b_3_6 + b_3_5
4. c_4_1 → b_2_1² + b_2_0·b_2_1 + b_2_0² + c_4_10
5. b_5_2 → b_2_0·b_3_0 + c_4_10·b_1_0
6. b_6_2 → b_3_2² + b_2_0·c_4_10
7. b_6_1 → b_3_2·b_3_5 + b_2_0·b_2_1² + b_2_0²·b_2_1 + b_2_1·c_4_10
8. b_7_6 → c_4_10·b_3_2
9. b_7_5 → c_4_10·b_3_5

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

1. b_2_0 → 0, an element of degree 2
2. b_3_1 → 0, an element of degree 3
3. b_3_0 → 0, an element of degree 3
4. c_4_1 → c_1_0⁴, an element of degree 4
5. b_5_2 → 0, an element of degree 5
6. b_6_2 → 0, an element of degree 6
7. b_6_1 → 0, an element of degree 6
8. b_7_6 → 0, an element of degree 7
9. b_7_5 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → c_1_1·c_1_2 + c_1_1², an element of degree 2
2. b_3_1 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
3. b_3_0 → 0, an element of degree 3
4. c_4_1 → c_1_2⁴ + c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_1·c_1_2²
      + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1²
      + c_1_0⁴, an element of degree 4
5. b_5_2 → c_1_1·c_1_2⁴ + c_1_1²·c_1_2³, an element of degree 5
6. b_6_2 → c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1²·c_1_2³ + c_1_0²·c_1_2⁴
      + c_1_0²·c_1_1·c_1_2³ + c_1_0²·c_1_1²·c_1_2² + c_1_0⁴·c_1_2², an element of degree 6
7. b_6_1 → c_1_1·c_1_2⁵ + c_1_1²·c_1_2⁴ + c_1_0·c_1_1²·c_1_2³ + c_1_0·c_1_1³·c_1_2²
      + c_1_0²·c_1_1·c_1_2³ + c_1_0²·c_1_1²·c_1_2² + c_1_0²·c_1_1³·c_1_2
      + c_1_0⁴·c_1_1·c_1_2, an element of degree 6
8. b_7_6 → 0, an element of degree 7
9. b_7_5 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
2. b_3_1 → 0, an element of degree 3
3. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
4. c_4_1 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
5. b_5_2 → 0, an element of degree 5
6. b_6_2 → 0, an element of degree 6
7. b_6_1 → 0, an element of degree 6
8. b_7_6 → 0, an element of degree 7
9. b_7_5 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → 0, an element of degree 2
2. b_3_1 → 0, an element of degree 3
3. b_3_0 → 0, an element of degree 3
4. c_4_1 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
5. b_5_2 → 0, an element of degree 5
6. b_6_2 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2
      + c_1_0²·c_1_2⁴ + c_1_0²·c_1_1²·c_1_2² + c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_2²
      + c_1_0⁴·c_1_1·c_1_2 + c_1_0⁴·c_1_1², an element of degree 6
7. b_6_1 → 0, an element of degree 6
8. b_7_6 → c_1_0·c_1_1²·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_1·c_1_2⁴
      + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7
9. b_7_5 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → 0, an element of degree 2
2. b_3_1 → 0, an element of degree 3
3. b_3_0 → 0, an element of degree 3
4. c_4_1 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
5. b_5_2 → 0, an element of degree 5
6. b_6_2 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2
      + c_1_0²·c_1_2⁴ + c_1_0²·c_1_1²·c_1_2² + c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_2²
      + c_1_0⁴·c_1_1·c_1_2 + c_1_0⁴·c_1_1², an element of degree 6
7. b_6_1 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2
      + c_1_0²·c_1_2⁴ + c_1_0²·c_1_1²·c_1_2² + c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_2²
      + c_1_0⁴·c_1_1·c_1_2 + c_1_0⁴·c_1_1², an element of degree 6
8. b_7_6 → c_1_0·c_1_1²·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_1·c_1_2⁴
      + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7
9. b_7_5 → c_1_0·c_1_1²·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_1·c_1_2⁴
      + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7

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

1. b_2_0 → c_1_2·c_1_3 + c_1_2² + c_1_1·c_1_2 + c_1_1² + c_1_0·c_1_1, an element of degree 2
2. b_3_1 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_2² + c_1_1²·c_1_2 + c_1_0·c_1_1²
      + c_1_0²·c_1_1, an element of degree 3
3. b_3_0 → 0, an element of degree 3
4. c_4_1 → c_1_3⁴ + c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_3²
      + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0·c_1_1·c_1_3²
      + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_1²·c_1_3 + c_1_0²·c_1_3²
      + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_3 + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + c_1_0³·c_1_1 + c_1_0⁴, an element of degree 4
5. b_5_2 → c_1_2·c_1_3⁴ + c_1_2²·c_1_3³ + c_1_1·c_1_2²·c_1_3² + c_1_1²·c_1_2·c_1_3²
      + c_1_0·c_1_1·c_1_2·c_1_3² + c_1_0·c_1_1·c_1_2²·c_1_3 + c_1_0·c_1_1²·c_1_2·c_1_3
      + c_1_0²·c_1_1·c_1_2·c_1_3 + c_1_0²·c_1_1·c_1_2² + c_1_0²·c_1_1²·c_1_2
      + c_1_0³·c_1_1² + c_1_0⁴·c_1_1, an element of degree 5
6. b_6_2 → c_1_0·c_1_2·c_1_3⁴ + c_1_0·c_1_2²·c_1_3³ + c_1_0·c_1_1·c_1_2²·c_1_3²
      + c_1_0·c_1_1²·c_1_2·c_1_3² + c_1_0²·c_1_3⁴ + c_1_0²·c_1_2·c_1_3³
      + c_1_0²·c_1_2²·c_1_3² + c_1_0²·c_1_1·c_1_2²·c_1_3 + c_1_0²·c_1_1²·c_1_3²
      + c_1_0²·c_1_1²·c_1_2·c_1_3 + c_1_0³·c_1_1·c_1_3² + c_1_0³·c_1_1²·c_1_3
      + c_1_0⁴·c_1_3² + c_1_0⁴·c_1_1·c_1_3, an element of degree 6
7. b_6_1 → c_1_2·c_1_3⁵ + c_1_2²·c_1_3⁴ + c_1_1·c_1_2²·c_1_3³ + c_1_1²·c_1_2·c_1_3³
      + c_1_0·c_1_2²·c_1_3³ + c_1_0·c_1_2³·c_1_3² + c_1_0·c_1_1·c_1_3⁴
      + c_1_0·c_1_1·c_1_2²·c_1_3² + c_1_0·c_1_1³·c_1_3² + c_1_0²·c_1_2·c_1_3³
      + c_1_0²·c_1_2²·c_1_3² + c_1_0²·c_1_2³·c_1_3 + c_1_0²·c_1_1²·c_1_3²
      + c_1_0²·c_1_1²·c_1_2·c_1_3 + c_1_0²·c_1_1³·c_1_3 + c_1_0³·c_1_1·c_1_3²
      + c_1_0³·c_1_1·c_1_2² + c_1_0³·c_1_1²·c_1_3 + c_1_0³·c_1_1²·c_1_2
      + c_1_0⁴·c_1_2·c_1_3 + c_1_0⁴·c_1_1² + c_1_0⁵·c_1_1, an element of degree 6
8. b_7_6 → 0, an element of degree 7
9. b_7_5 → 0, an element of degree 7

About the group

Ring generators

Ring relations

Completion information

Restriction maps