Mod-11-Cohomology of MathieuGroup(22), a group of order 443520

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • MathieuGroup(22) is a group of order 443520.
  • The group order factors as 27 · 32 · 5 · 7 · 11.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22),(1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),(1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20)]).
  • 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

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(55,1); GF(11)).

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].

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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

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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


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