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Mod-2-Cohomology of SymplecticGroup(4,3), a group of order 51840

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• SymplecticGroup(4,3) is a group of order 51840.
• The group order factors as 2⁷ · 3⁴ · 5.
• The group is defined by Group([(1,2)(3,5)(4,7)(6,10)(8,12)(9,13)(11,16)(14,20)(15,21)(17,24)(18,25)(19,27)(23,31)(28,32)(29,37)(30,39)(33,38)(34,43)(35,45)(41,49)(46,54)(47,51)(48,57)(50,53)(52,61)(55,56)(58,67)(59,60)(62,70)(65,71)(66,72)(68,74)(69,75)(73,76)(77,79)(78,80),(1,3,6)(2,4,8)(5,9,14,12,18,26,34,44,25)(7,11,17,10,15,22,30,40,21)(13,19,28,36,47,56,20,29,38)(16,23,32,42,51,60,24,33,37)(27,35,46,55,65,67,73,57,66)(31,41,50,59,68,70,76,61,69)(39,48,58)(43,52,62)(45,53,63)(49,54,64)(71,77,75)(72,74,78)]).
• It is non-abelian.
• It has 2-Rank 2.
• 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 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) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6)

  ( − 1  +  t)2 · (1  +  t2)2 · (1  +  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 4 minimal generators of maximal degree 8:

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. c_8_1, 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. a_3_0²
2. a_7_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_1, an element of degree 8
  2. b_4_0, an element of degree 4
• A Duflot regular sequence is given by c_8_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(Syl2(Sp4(3)); GF(2))

1. a_3_0 → a_2_4·a_1_2 + a_1_2³ + a_1_1³
2. b_4_0 → b_1_0⁴ + b_4_8
3. a_7_0 → a_2_4·a_5_10 + a_2_4·a_2_5·b_1_0³ + a_1_2²·a_5_10 + a_1_1²·a_5_10 + a_1_1²·a_5_9
      + b_4_8·a_1_2³ + b_4_8·a_1_1·a_1_2²
4. c_8_1 → b_4_8² + a_2_4·a_2_5·b_1_0⁴ + a_2_4²·b_1_0⁴ + a_2_4²·a_2_5·b_1_0²
      + a_1_2³·a_5_10 + a_1_1³·a_5_10 + a_1_1³·a_5_9 + b_4_8·a_1_1²·a_1_2²
      + b_4_8·a_1_1³·a_1_2 + c_8_15

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. c_8_1 → c_1_0⁸, an element of degree 8

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 → c_1_1⁴, an element of degree 4
3. a_7_0 → 0, an element of degree 7
4. c_8_1 → c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

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 → c_1_1⁴, an element of degree 4
3. a_7_0 → 0, an element of degree 7
4. c_8_1 → 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