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Mod-3-Cohomology of MathieuGroup(9), a group of order 72

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• MathieuGroup(9) is a group of order 72.
• The group order factors as 2³ · 3².
• The group is defined by Group([(1,4,9,8)(2,5,3,6),(1,6,5,2)(3,7,9,8)]).
• It is non-abelian.
• It has 3-Rank 2.
• The centre of a Sylow 3-subgroup has rank 2.
• Its Sylow 3-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 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  1  −  2·t  +  4·t2  −  5·t3  +  5·t4  −  5·t5  +  4·t6  −  2·t7  +  t8

  ( − 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 8 minimal generators of maximal degree 12:

1. a_2_0, a nilpotent element of degree 2
2. a_3_0, a nilpotent element of degree 3
3. a_7_1, a nilpotent element of degree 7
4. a_7_0, a nilpotent element of degree 7
5. c_8_1, a Duflot element of degree 8
6. c_8_0, a Duflot element of degree 8
7. a_11_0, a nilpotent element of degree 11
8. c_12_0, a Duflot element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 4 "obvious" relations:
   a_3_0², a_7_0², a_7_1², a_11_0²

Apart from that, there are 16 minimal relations of maximal degree 24:

1. a_2_0²
2. a_2_0·a_3_0
3. a_2_0·a_7_0
4. a_2_0·a_7_1
5. a_3_0·a_7_0 − a_2_0·c_8_0
6. a_3_0·a_7_1 − a_2_0·c_8_1
7. a_2_0·a_11_0
8. a_3_0·a_11_0 − a_2_0·c_12_0
9. a_7_0·a_7_1 + a_2_0·c_12_0
10. c_12_0·a_3_0 + c_8_1·a_7_0 − c_8_0·a_7_1
11. a_7_0·a_11_0 − a_2_0·c_8_0·c_8_1
12. a_7_1·a_11_0 − a_2_0·c_8_0²
13. c_12_0·a_7_0 − c_8_0·a_11_0 − c_8_0·c_8_1·a_3_0
14. c_12_0·a_7_1 − c_8_1·a_11_0 − c_8_0²·a_3_0
15. c_12_0·a_11_0 + c_8_0·c_8_1·a_7_1 − c_8_0²·a_7_0
16. c_12_0² + c_8_0·c_8_1² − c_8_0³

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 24 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. c_8_0, an element of degree 8
• The above filter regular HSOP forms a Duflot regular sequence.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 14].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(9,2); GF(3))

1. a_2_0 → a_1_0·a_1_1
2. a_3_0 → c_2_2·a_1_0 − c_2_1·a_1_1
3. a_7_1 → c_2_2³·a_1_0 − c_2_1³·a_1_1
4. a_7_0 → c_2_2³·a_1_1 + c_2_1·c_2_2²·a_1_0 + c_2_1²·c_2_2·a_1_1 + c_2_1³·a_1_0
5. c_8_1 → c_2_1·c_2_2³ − c_2_1³·c_2_2
6. c_8_0 → c_2_2⁴ − c_2_1²·c_2_2² + c_2_1⁴
7. a_11_0 → c_2_2⁵·a_1_1 + c_2_1²·c_2_2³·a_1_1 + c_2_1³·c_2_2²·a_1_0 + c_2_1⁵·a_1_0
8. c_12_0 → c_2_2⁶ + c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

1. a_2_0 → a_1_0·a_1_1, an element of degree 2
2. a_3_0 → c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
3. a_7_1 → c_2_2³·a_1_0 − c_2_1³·a_1_1, an element of degree 7
4. a_7_0 → c_2_2³·a_1_1 + c_2_1·c_2_2²·a_1_0 + c_2_1²·c_2_2·a_1_1 + c_2_1³·a_1_0, an element of degree 7
5. c_8_1 → c_2_1·c_2_2³ − c_2_1³·c_2_2, an element of degree 8
6. c_8_0 → c_2_2⁴ − c_2_1²·c_2_2² + c_2_1⁴, an element of degree 8
7. a_11_0 → c_2_2⁵·a_1_1 + c_2_1²·c_2_2³·a_1_1 + c_2_1³·c_2_2²·a_1_0 + c_2_1⁵·a_1_0, an element of degree 11
8. c_12_0 → c_2_2⁶ + c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶, an element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps