Mathematics:

Cohomology
→Theory
→Implementation

Jena:

Faculty

David Green

External links:

Singular

Gap

Mod-2-Cohomology of MathieuGroup(10), a group of order 720

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• MathieuGroup(10) is a group of order 720.
• The group order factors as 2⁴ · 3² · 5.
• The group is defined by Group([(1,9,6,7,5)(2,10,3,8,4),(1,10,7,8)(2,9,4,6)]).
• It is non-abelian.
• It has 2-Rank 2.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1)·( − 1  −  t2  +  t3)

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

• The a-invariants are -∞,-2,-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 5:

1. a_1_0, a nilpotent element of degree 1
2. b_3_0, an element of degree 3
3. c_4_0, a Duflot element of degree 4
4. b_5_0, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 4 minimal relations of maximal degree 10:

1. a_1_0³
2. a_1_0·b_3_0
3. a_1_0·b_5_0
4. b_5_0² + c_4_0·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 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_3_0, an element of degree 3
• A Duflot regular sequence is given by c_4_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, 2, 5].
• We found that there exists some HSOP over a finite extension field, in degrees 4,3.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SD16; GF(2))

1. a_1_0 → a_1_0
2. b_3_0 → b_3_1
3. c_4_0 → b_1_1⁴ + c_4_2
4. b_5_0 → b_1_1²·b_3_1 + c_4_2·b_1_1

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

1. a_1_0 → 0, an element of degree 1
2. b_3_0 → 0, an element of degree 3
3. c_4_0 → c_1_0⁴, an element of degree 4
4. b_5_0 → 0, an element of degree 5

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

1. a_1_0 → 0, an element of degree 1
2. b_3_0 → c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
3. c_4_0 → c_1_1⁴ + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
4. b_5_0 → c_1_0·c_1_1⁴ + c_1_0⁴·c_1_1, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps