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Mod-2-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 2-Rank 1.
- 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 1.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(Q8; GF(2)).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 + t) · (1 + t2) - The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Benson 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 3 minimal generators of maximal degree 4:
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a_1_0 , a nilpotent element of degree 1 -
a_1_1 , a nilpotent element of degree 1 -
c_4_0 , a Duflot element of degree 4
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Ring relations
There are 2 minimal relations of maximal degree 3:
-
a_1_12 + a_1_0·a_1_1 + a_1_02 -
a_1_03
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Data used for the Benson test
- We proved completion in degree 4 using the Benson 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:
c_4_0 , an element of degree 4
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, 3].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(Q8; GF(2))
-
a_1_0 →a_1_0 -
a_1_1 →a_1_1 -
c_4_0 →c_4_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
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a_1_0 →0 , an element of degree 1 -
a_1_1 →0 , an element of degree 1 -
c_4_0 →c_1_04 , an element of degree 4
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
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Simon King Department of Mathematics and Computer Science Friedrich-Schiller-Universität Jena 07737 Jena GERMANY
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