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Mod-2-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 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  −  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. c_4_0, a Duflot element of degree 4
3. a_5_0, a nilpotent element of degree 5
4. b_6_0, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 minimal relations of maximal degree 10:

1. a_3_0²
2. a_5_0²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 10 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_4_0, an element of degree 4
  2. b_6_0, an element of degree 6
• A Duflot regular sequence is given by c_4_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 8].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(96,67); GF(2))

1. a_3_0 → a_3_0
2. c_4_0 → b_2_0² + a_1_0·a_3_0 + c_4_2
3. a_5_0 → b_2_0·a_3_0 + c_4_2·a_1_0
4. b_6_0 → b_2_0·a_1_0·a_3_0 + b_2_0·c_4_2

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. c_4_0 → c_1_0⁴, an element of degree 4
3. a_5_0 → 0, an element of degree 5
4. b_6_0 → 0, an element of degree 6

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. c_4_0 → c_1_1⁴ + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. a_5_0 → 0, an element of degree 5
4. b_6_0 → c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_1², an element of degree 6

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. c_4_0 → c_1_1⁴ + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. a_5_0 → 0, an element of degree 5
4. b_6_0 → c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_1², an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps