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David Green

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Mod-2-Cohomology of MathieuGroup(11), a group of order 7920

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• MathieuGroup(11) is a group of order 7920.
• The group order factors as 2⁴ · 3² · 5 · 11.
• The group is defined by Group([(1,2,3,4,5,6,7,8,9,10,11),(3,7,11,8)(4,10,5,6)]).
• 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 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 3 minimal generators of maximal degree 5:

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

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There is one minimal relation of degree 10:

1. 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, -1, 5].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(48,29); GF(2))

1. b_3_0 → b_3_0
2. c_4_0 → b_1_0⁴ + c_4_2
3. b_5_0 → b_1_0²·b_3_0 + c_4_2·b_1_0

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. c_4_0 → c_1_0⁴, an element of degree 4
3. 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. b_3_0 → c_1_0·c_1_1² + c_1_0²·c_1_1, 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. 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