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Mod-11-Cohomology of MathieuGroup(12), a group of order 95040

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• MathieuGroup(12) is a group of order 95040.
• The group order factors as 2⁶ · 3³ · 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),(1,12)(2,11)(3,6)(4,8)(5,9)(7,10)]).
• It is non-abelian.
• It has 11-Rank 1.
• The centre of a Sylow 11-subgroup has rank 1.
• Its Sylow 11-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

General information

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

  ( − 1)·((1  −  t  +  t2) · (1  −  t3  +  t6))

  ( − 1  +  t) · (1  −  t  +  t2  −  t3  +  t4) · (1  +  t  +  t2  +  t3  +  t4)

• The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
• The filter degree type of any filter regular HSOP is [-1, -1].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 2 minimal generators of maximal degree 10:

1. a_9_0, a nilpotent element of degree 9
2. c_10_0, a Duflot element of degree 10

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There is one "obvious" relation:
   a_9_0²

Apart from that, there are no relations.

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_10_0, an element of degree 10
• The above filter regular HSOP forms a Duflot regular sequence.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, 9].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(55,1); GF(11))

1. a_9_0 → a_9_0
2. c_10_0 → c_10_0

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

1. a_9_0 → c_2_0⁴·a_1_0, an element of degree 9
2. c_10_0 → c_2_0⁵, an element of degree 10

About the group

Ring generators

Ring relations

Completion information

Restriction maps