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Mod-3-Cohomology of U3(3), a group of order 6048

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• U3(3) is a group of order 6048.
• The group order factors as 2⁵ · 3³ · 7.
• The group is defined by Group([(2,3)(4,6)(5,8)(7,11)(9,13)(10,15)(12,14)(16,20)(17,22)(18,23)(24,27)(25,28),(1,2,4,7,12,17)(3,5,9,14,19,22)(6,10,13,18,24,23)(8,11,16,21,26,28)(20,25,27)]).
• It is non-abelian.
• It has 3-Rank 2.
• The centre of a Sylow 3-subgroup has rank 1.
• Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

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

  1  −  2·t  +  2·t2  −  t3  +  2·t4  −  3·t5  +  2·t6  −  t7  +  2·t8  −  2·t9  +  t10

  ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4)

• The a-invariants are -∞,-∞,-2. 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, -2].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 8 minimal generators of maximal degree 12:

1. a_3_0, a nilpotent element of degree 3
2. a_4_1, a nilpotent element of degree 4
3. b_4_0, an element of degree 4
4. a_5_0, a nilpotent element of degree 5
5. a_9_1, a nilpotent element of degree 9
6. b_10_0, an element of degree 10
7. a_11_1, a nilpotent element of degree 11
8. c_12_1, a Duflot element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 4 "obvious" relations:
   a_3_0², a_5_0², a_9_1², a_11_1²

Apart from that, there are 16 minimal relations of maximal degree 21:

1. a_4_1·a_3_0
2. a_4_1²
3. a_4_1·b_4_0 − a_3_0·a_5_0
4. a_4_1·a_5_0
5. a_3_0·a_9_1
6. a_4_1·a_9_1
7. b_10_0·a_3_0 − b_4_0·a_9_1
8. a_5_0·a_9_1 − a_3_0·a_11_1
9. a_4_1·b_10_0 + a_3_0·a_11_1
10. a_4_1·a_11_1
11. b_10_0·a_5_0 + b_4_0·a_11_1
12. a_5_0·a_11_1
13. b_10_0·a_9_1 + b_4_0·c_12_1·a_3_0
14. a_9_1·a_11_1 − c_12_1·a_3_0·a_5_0
15. b_10_0² + b_4_0²·c_12_1
16. b_10_0·a_11_1 − b_4_0·c_12_1·a_5_0

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 21 using the Hilbert-Poincaré criterion.
• 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_12_1, an element of degree 12
  2. b_4_0, an element of degree 4
• A Duflot regular sequence is given by c_12_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 14].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(216,86); GF(3))

1. a_3_0 → a_3_0
2. a_4_1 → a_4_1
3. b_4_0 → b_4_0
4. a_5_0 → a_5_0
5. a_9_1 → a_9_1
6. b_10_0 → b_10_0
7. a_11_1 → a_11_1
8. c_12_1 → c_12_1

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. a_4_1 → 0, an element of degree 4
3. b_4_0 → 0, an element of degree 4
4. a_5_0 → 0, an element of degree 5
5. a_9_1 → 0, an element of degree 9
6. b_10_0 → 0, an element of degree 10
7. a_11_1 → 0, an element of degree 11
8. c_12_1 →  − c_2_0⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 →  − c_2_1·c_2_2³·a_1_1 + c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 →  − c_2_1·c_2_2⁴ + c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → 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_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 → c_2_1·c_2_2³·a_1_1 − c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 → c_2_1·c_2_2⁴ − c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → 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_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 → c_2_1·c_2_2³·a_1_1 − c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 → c_2_1·c_2_2⁴ − c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → 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_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 →  − c_2_1·c_2_2³·a_1_1 + c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 →  − c_2_1·c_2_2⁴ + c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → 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_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps