Mod-23-Cohomology of MathieuGroup(23), a group of order 10200960

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General information on the group

  • MathieuGroup(23) is a group of order 10200960.
  • The group order factors as 27 · 32 · 5 · 7 · 11 · 23.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23),(3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16)]).
  • It is non-abelian.
  • It has 23-Rank 1.
  • The centre of a Sylow 23-subgroup has rank 1.
  • Its Sylow 23-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(253,1); GF(23)).

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  +  t2  −  t3  +  t4  −  t5  +  t6) · (1  +  t  −  t3  −  t4  +  t6  −  t8  −  t9  +  t11  +  t12))
    ( − 1  +  t) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10) · (1  +  t  +  t2  +  t3  +  t4  +  t5  +  t6  +  t7  +  t8  +  t9  +  t10)
  • 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 22:

  1. a_21_0, a nilpotent element of degree 21
  2. c_22_0, a Duflot element of degree 22

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Ring relations

There is one "obvious" relation:
   a_21_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 22 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_22_0, an element of degree 22
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 21].

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(253,1); GF(23))

  1. a_21_0 → a_21_0
  2. c_22_0 → c_22_0

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

  1. a_21_0 → c_2_010·a_1_0, an element of degree 21
  2. c_22_0 → c_2_011, an element of degree 22

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