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Mod-3-Cohomology of SymmetricGroup(9), a group of order 362880

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• SymmetricGroup(9) is a group of order 362880.
• The group order factors as 2⁷ · 3⁴ · 5 · 7.
• The group is defined by Group([(1,2,3,4,5,6,7,8,9),(1,2)]).
• It is non-abelian.
• It has 3-Rank 3.
• The centre of a Sylow 3-subgroup has rank 1.
• Its Sylow 3-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1)·(1  −  3·t  +  6·t2  −  9·t3  +  12·t4  −  15·t5  +  18·t6  −  20·t7  +  22·t8  −  24·t9  +  28·t10  −  31·t11  +  33·t12  −  34·t13  +  34·t14  −  32·t15  +  29·t16  −  25·t17  +  22·t18  −  20·t19  +  18·t20  −  15·t21  +  12·t22  −  9·t23  +  6·t24  −  3·t25  +  t26)

  ( − 1  +  t)3 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)3 · (1  −  t2  +  t4) · (1  +  t4) · (1  +  t8)

• The a-invariants are -∞,-∞,-12,-3. 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, -3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 10 minimal generators of maximal degree 16:

1. a_3_0, a nilpotent element of degree 3
2. b_4_0, an element of degree 4
3. a_7_0, a nilpotent element of degree 7
4. b_8_0, an element of degree 8
5. a_10_0, a nilpotent element of degree 10
6. a_11_1, a nilpotent element of degree 11
7. a_11_0, a nilpotent element of degree 11
8. c_12_0, a Duflot element of degree 12
9. a_15_0, a nilpotent element of degree 15
10. b_16_4, an element of degree 16

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 5 "obvious" relations:
   a_3_0², a_7_0², a_11_0², a_11_1², a_15_0²

Apart from that, there are 24 minimal relations of maximal degree 27:

1. a_10_0·a_3_0
2. a_3_0·a_11_0
3. b_4_0·a_10_0
4. b_4_0·a_11_0
5. a_10_0·a_7_0
6. a_3_0·a_15_0
7. a_7_0·a_11_0
8. b_8_0·a_10_0
9. b_4_0·a_15_0
10. b_8_0·a_11_0
11. b_16_4·a_3_0
12. a_10_0²
13. b_4_0·b_16_4
14. a_10_0·a_11_0
15. a_10_0·a_11_1
16. a_7_0·a_15_0
17. a_11_0·a_11_1
18. b_8_0·a_15_0
19. b_16_4·a_7_0
20. b_8_0·b_16_4
21. a_10_0·a_15_0
22. a_11_1·a_15_0
23. a_10_0·b_16_4 + a_11_0·a_15_0
24. b_16_4·a_11_1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 34 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 27 and the last generator in degree 16.
• The following is a filter regular homogeneous system of parameters:
  1.  − b_4_0·b_8_0⁴ − b_4_0³·b_8_0³ − b_4_0⁹ + b_8_0³·c_12_0 + b_4_0²·b_8_0²·c_12_0
        + c_12_0³, an element of degree 36
  2. b_16_4³ − b_8_0⁶ + b_4_0²·b_8_0⁵ + b_4_0⁴·b_8_0⁴ + b_4_0⁶·b_8_0³
        + b_4_0³·b_8_0³·c_12_0 − b_4_0⁵·b_8_0²·c_12_0 − b_8_0³·c_12_0²
        + b_4_0²·b_8_0²·c_12_0² − b_4_0³·c_12_0³, an element of degree 48
  3. b_8_0, an element of degree 8
• A Duflot regular sequence is given by c_12_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 72, 89].
• Modifying the above filter regular HSOP, we obtained the following parameters:
1. b_4_0³ + c_12_0, an element of degree 12
2. b_16_4 + b_4_0·c_12_0, an element of degree 16
3. b_8_0, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(324,39); GF(3))

1. a_3_0 → a_3_0
2. b_4_0 → b_4_0
3. a_7_0 → a_7_6 + a_7_1 + b_4_1·a_3_0 + b_4_0·a_3_1
4. b_8_0 → b_8_4 − b_4_0·b_4_1
5. a_10_0 → a_10_6
6. a_11_1 → a_11_9 − b_4_1·a_7_1 + b_4_1²·a_3_0 − b_4_0·b_4_1·a_3_1
7. a_11_0 → a_11_13
8. c_12_0 → b_4_2³ + b_4_1·b_8_4 + b_4_1³ + b_4_0·b_4_1² + c_12_7
9. a_15_0 → b_4_2·a_11_13 + c_12_7·a_3_2
10. b_16_4 → b_4_2·c_12_7

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

1. a_3_0 → 0, an element of degree 3
2. b_4_0 → 0, an element of degree 4
3. a_7_0 → 0, an element of degree 7
4. b_8_0 → 0, an element of degree 8
5. a_10_0 → 0, an element of degree 10
6. a_11_1 → 0, an element of degree 11
7. a_11_0 → 0, an element of degree 11
8. c_12_0 → c_2_0⁶, an element of degree 12
9. a_15_0 → 0, an element of degree 15
10. b_16_4 → 0, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. a_3_0 → 0, an element of degree 3
2. b_4_0 → 0, an element of degree 4
3. a_7_0 → 0, an element of degree 7
4. b_8_0 → 0, an element of degree 8
5. a_10_0 →  − c_2_1·c_2_2³·a_1_0·a_1_1 + c_2_1³·c_2_2·a_1_0·a_1_1, an element of degree 10
6. a_11_1 → 0, an element of degree 11
7. a_11_0 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
8. c_12_0 → c_2_2⁶ + c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶, an element of degree 12
9. a_15_0 → c_2_1·c_2_2⁶·a_1_0 − c_2_1³·c_2_2⁴·a_1_0 − c_2_1⁴·c_2_2³·a_1_1
      + c_2_1⁶·c_2_2·a_1_1, an element of degree 15
10. b_16_4 → c_2_1²·c_2_2⁶ + c_2_1⁴·c_2_2⁴ + c_2_1⁶·c_2_2², an element of degree 16

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

1. a_3_0 →  − c_2_5·a_1_1 + c_2_5·a_1_0 − c_2_4·a_1_2 − c_2_4·a_1_1 + c_2_3·a_1_2, an element of degree 3
2. b_4_0 → c_2_4·c_2_5 − c_2_4² − c_2_3·c_2_5, an element of degree 4
3. a_7_0 →  − c_2_5³·a_1_1 + c_2_5³·a_1_0 − c_2_4·c_2_5²·a_1_1 + c_2_4·c_2_5²·a_1_0
      − c_2_4²·c_2_5·a_1_2 − c_2_4²·c_2_5·a_1_0 + c_2_3·c_2_5²·a_1_1 − c_2_3·c_2_5²·a_1_0
      − c_2_3·c_2_4·c_2_5·a_1_2 + c_2_3·c_2_4·c_2_5·a_1_1 − c_2_3·c_2_4²·a_1_2
      − c_2_3²·c_2_5·a_1_2 + c_2_3³·a_1_2, an element of degree 7
4. b_8_0 →  − c_2_4·c_2_5³ + c_2_4²·c_2_5² + c_2_3·c_2_5³ + c_2_3·c_2_4·c_2_5²
      − c_2_3·c_2_4²·c_2_5 + c_2_3²·c_2_5² + c_2_3³·c_2_5, an element of degree 8
5. a_10_0 → 0, an element of degree 10
6. a_11_1 →  − c_2_5⁵·a_1_1 + c_2_5⁵·a_1_0 + c_2_4·c_2_5⁴·a_1_2 − c_2_4·c_2_5⁴·a_1_1
      − c_2_4·c_2_5⁴·a_1_0 + c_2_4²·c_2_5³·a_1_2 + c_2_4²·c_2_5³·a_1_0
      − c_2_3·c_2_5⁴·a_1_2 − c_2_3·c_2_5⁴·a_1_1 + c_2_3·c_2_5⁴·a_1_0
      − c_2_3·c_2_4·c_2_5³·a_1_2 − c_2_3·c_2_4·c_2_5³·a_1_1 + c_2_3·c_2_4²·c_2_5²·a_1_0
      + c_2_3·c_2_4³·c_2_5·a_1_0 + c_2_3·c_2_4⁴·a_1_0 − c_2_3²·c_2_5³·a_1_2
      + c_2_3²·c_2_4·c_2_5²·a_1_1 + c_2_3²·c_2_4²·c_2_5·a_1_2 − c_2_3²·c_2_4³·a_1_2
      − c_2_3²·c_2_4³·a_1_1 + c_2_3³·c_2_5²·a_1_1 − c_2_3³·c_2_5²·a_1_0
      − c_2_3³·c_2_4·c_2_5·a_1_2 + c_2_3³·c_2_4·c_2_5·a_1_1 + c_2_3³·c_2_4·c_2_5·a_1_0
      − c_2_3³·c_2_4²·a_1_2 − c_2_3³·c_2_4²·a_1_0 + c_2_3⁴·c_2_5·a_1_2
      + c_2_3⁴·c_2_5·a_1_1 − c_2_3⁴·c_2_5·a_1_0 + c_2_3⁴·c_2_4·a_1_2 + c_2_3⁴·c_2_4·a_1_1
      + c_2_3⁵·a_1_2, an element of degree 11
7. a_11_0 → 0, an element of degree 11
8. c_12_0 → c_2_5⁶ + c_2_4·c_2_5⁵ − c_2_4²·c_2_5⁴ − c_2_3·c_2_5⁵ + c_2_3·c_2_4·c_2_5⁴
      − c_2_3·c_2_4²·c_2_5³ + c_2_3²·c_2_5⁴ + c_2_3²·c_2_4²·c_2_5²
      + c_2_3²·c_2_4³·c_2_5 + c_2_3²·c_2_4⁴ + c_2_3³·c_2_5³ − c_2_3³·c_2_4·c_2_5²
      + c_2_3³·c_2_4²·c_2_5 + c_2_3⁴·c_2_5² − c_2_3⁴·c_2_4·c_2_5 + c_2_3⁴·c_2_4²
      − c_2_3⁵·c_2_5 + c_2_3⁶, an element of degree 12
9. a_15_0 → 0, an element of degree 15
10. b_16_4 → 0, an element of degree 16

About the group

Ring generators

Ring relations

Completion information

Restriction maps