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Mod-3-Cohomology of SL(3,4), a group of order 60480

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• SL(3,4) is a group of order 60480.
• The group order factors as 2⁶ · 3³ · 5 · 7.
• The group is defined by Group([(2,3,5)(4,7,6)(8,11,17)(9,13,21)(10,15,25)(12,19,32)(14,23,39)(16,27,46)(18,30,51)(20,34,52)(22,37,55)(24,41,56)(26,44,50)(28,48,59)(29,45,47)(31,33,36)(35,54,63)(38,40,43)(42,57,61)(49,53,62),(1,2,4,8,12,20,35)(3,6,9,14,24,42,58)(5,7,10,16,28,49,60)(11,18,31,23,40,56,62)(13,22,38,27,47,59,63)(15,26,45,19,33,52,61)(17,29,50,46,37,48,57)(21,36,51,32,44,34,53)(25,43,55,39,30,41,54)]).
• 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  −  t  +  t2  −  t3  +  t4

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

• 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 4 minimal generators of maximal degree 6:

1. a_3_0, a nilpotent element of degree 3
2. b_4_0, an element of degree 4
3. a_5_0, a nilpotent element of degree 5
4. c_6_0, a Duflot element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 "obvious" relations:
   a_3_0², a_5_0²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 8 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 0 and the last generator in degree 6.
• The following is a filter regular homogeneous system of parameters:
  1. c_6_0, an element of degree 6
  2. b_4_0, an element of degree 4
• A Duflot regular sequence is given by c_6_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 8].
• We found that there exists some HSOP over a finite extension field, in degrees 6,4.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(E27; GF(3))

1. a_3_0 → b_2_3·a_1_1 − b_2_1·a_1_1 + b_2_0·a_1_0
2. b_4_0 → b_2_3² − b_2_0·b_2_2 − b_2_0·b_2_1 + b_2_0² + a_1_0·a_3_5 − a_1_0·a_3_4
3. a_5_0 → b_2_3·a_3_5 − b_2_1·a_3_5 + b_2_0·a_3_4 + b_2_0·b_2_1·a_1_1 − b_2_0²·a_1_1
4. c_6_0 → b_2_0²·b_2_2 − b_2_0·a_1_1·a_3_5 − b_2_0·a_1_0·a_3_5 − b_2_0·a_1_0·a_3_4 − c_6_8

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_5_0 → 0, an element of degree 5
4. c_6_0 → c_2_0³, an element of degree 6

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. b_4_0 → c_2_2², an element of degree 4
3. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
4. c_6_0 →  − c_2_1·c_2_2² + c_2_1³, an element of degree 6

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. b_4_0 → c_2_2², an element of degree 4
3. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
4. c_6_0 →  − c_2_1·c_2_2² + c_2_1³, an element of degree 6

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. b_4_0 → c_2_2², an element of degree 4
3. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
4. c_6_0 →  − c_2_1·c_2_2² + c_2_1³, an element of degree 6

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. b_4_0 → c_2_2², an element of degree 4
3. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
4. c_6_0 →  − c_2_1·c_2_2² + c_2_1³, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps