Mathematics:

Cohomology
→Theory
→Implementation

Jena:

Faculty

David Green

External links:

Singular

Gap

Mod-2-Cohomology of SL(4,3), a group of order 12130560

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• SL(4,3) is a group of order 12130560.
• The group order factors as 2⁸ · 3⁶ · 5 · 13.
• The group is defined by Group([(2,3,5)(4,7,12)(6,10,17)(8,14,23)(11,19,31)(16,26,40)(18,29,44)(21,32,45)(24,37,52)(25,33,42)(35,48,59)(36,50,51)(38,46,56)(39,49,41)(47,55,60)(53,54,58)(65,70,69)(74,76,77),(1,2,4,8,15,25,39,54)(3,6,11,20,33,47,52,63)(5,9,16,27,42,57,65,71)(7,13,22,35,49,61,68,74)(10,18,30,40,55,50,62,69)(12,21,34,26,41,56,64,70)(14,24,38,45,58,19,32,46)(17,28,43,48,60,67,73,76)(23,36,51,37,53,44,29,31)(59,66,72,75,77,78,79,80)]).
• It is non-abelian.
• It has 2-Rank 3.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

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

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

  ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t2)2 · (1  +  t4)

• The a-invariants are -∞,-∞,-∞,-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 5 minimal generators of maximal degree 8:

1. b_3_0, an element of degree 3
2. b_4_0, an element of degree 4
3. b_5_0, an element of degree 5
4. b_7_0, an element of degree 7
5. c_8_2, a Duflot element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 minimal relations of maximal degree 14:

1. b_5_0² + b_4_0·b_3_0²
2. b_7_0² + c_8_2·b_3_0²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 14 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_8_2, an element of degree 8
  2. b_4_0, an element of degree 4
  3. b_3_0, an element of degree 3
• A Duflot regular sequence is given by c_8_2.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 12].
• We found that there exists some HSOP over a finite extension field, in degrees 8,4,3.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(256,6671); GF(2))

1. b_3_0 → b_3_8 + b_1_1·b_1_2² + b_1_1²·b_1_2 + a_2_4·a_1_0 + a_1_0³
2. b_4_0 → b_1_2⁴ + b_1_1²·b_1_2² + b_1_1⁴ + b_4_11
3. b_5_0 → b_1_1·b_1_2⁴ + b_1_1²·b_3_8 + b_1_1⁴·b_1_2 + b_4_11·b_1_1
4. b_7_0 → b_7_21 + b_1_1⁴·b_3_8 + b_6_17·b_1_1 + b_4_11·b_1_1³ + a_1_0²·a_5_12
5. c_8_2 → b_6_17·b_1_2² + b_4_11² + a_2_5·b_1_2⁶ + a_2_4·b_1_1⁶ + a_2_4²·b_1_1⁴
      + a_1_0³·a_5_12 + c_8_25

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

1. b_3_0 → 0, an element of degree 3
2. b_4_0 → 0, an element of degree 4
3. b_5_0 → 0, an element of degree 5
4. b_7_0 → 0, an element of degree 7
5. c_8_2 → c_1_0⁸, an element of degree 8

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

1. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
2. b_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
3. b_5_0 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2, an element of degree 5
4. b_7_0 → 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
5. c_8_2 → c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1⁴·c_1_2² + c_1_0⁴·c_1_2⁴
      + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

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

1. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
2. b_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
3. b_5_0 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2, an element of degree 5
4. b_7_0 → 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
5. c_8_2 → c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1⁴·c_1_2² + c_1_0⁴·c_1_2⁴
      + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps