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:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. 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:

  1. a_1_12 + a_1_0·a_1_1 + a_1_02
  2. 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:
    1. 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))

  1. a_1_0a_1_0
  2. a_1_1a_1_1
  3. c_4_0c_4_0

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

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. c_4_0c_1_04, an element of degree 4


About the group Ring generators Ring relations Completion information Restriction maps




Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2



Last change: 14.12.2010