Mod-3-Cohomology of MathieuGroup(10), a group of order 720

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General information on the group

  • MathieuGroup(10) is a group of order 720.
  • The group order factors as 24 · 32 · 5.
  • The group is defined by Group([(1,9,6,7,5)(2,10,3,8,4),(1,10,7,8)(2,9,4,6)]).
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 2.
  • Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(72,41); GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1  −  2·t  +  4·t2  −  5·t3  +  5·t4  −  5·t5  +  4·t6  −  2·t7  +  t8
    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 12:

  1. a_2_0, a nilpotent element of degree 2
  2. a_3_0, a nilpotent element of degree 3
  3. a_7_0, a nilpotent element of degree 7
  4. a_7_1, a nilpotent element of degree 7
  5. c_8_0, a Duflot element of degree 8
  6. c_8_1, a Duflot element of degree 8
  7. a_11_0, a nilpotent element of degree 11
  8. c_12_0, a Duflot element of degree 12

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Ring relations

There are 4 "obvious" relations:
   a_3_02, a_7_02, a_7_12, a_11_02

Apart from that, there are 16 minimal relations of maximal degree 24:

  1. a_2_02
  2. a_2_0·a_3_0
  3. a_2_0·a_7_0
  4. a_2_0·a_7_1
  5. a_3_0·a_7_0 − a_2_0·c_8_0
  6. a_3_0·a_7_1 − a_2_0·c_8_1
  7. a_2_0·a_11_0
  8. a_3_0·a_11_0 − a_2_0·c_12_0
  9. a_7_0·a_7_1 + a_2_0·c_12_0
  10. c_12_0·a_3_0 + c_8_1·a_7_0 − c_8_0·a_7_1
  11. a_7_0·a_11_0 − a_2_0·c_8_0·c_8_1
  12. a_7_1·a_11_0 − a_2_0·c_8_02
  13. c_12_0·a_7_0 − c_8_0·a_11_0 − c_8_0·c_8_1·a_3_0
  14. c_12_0·a_7_1 − c_8_1·a_11_0 − c_8_02·a_3_0
  15. c_12_0·a_11_0 + c_8_0·c_8_1·a_7_1 − c_8_02·a_7_0
  16. c_12_02 + c_8_0·c_8_12 − c_8_03

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 24 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_8_1, an element of degree 8
    2. c_8_0, an element of degree 8
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 14].

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(72,41); GF(3))

  1. a_2_0 → a_2_0
  2. a_3_0 → a_3_0
  3. a_7_0 → a_7_0
  4. a_7_1 → a_7_1
  5. c_8_0 → c_8_0
  6. c_8_1 → c_8_1
  7. a_11_0 → a_11_0
  8. c_12_0 → c_12_0

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_2_0 → a_1_0·a_1_1, an element of degree 2
  2. a_3_0 → c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
  3. a_7_0 → c_2_23·a_1_1 + c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1 + c_2_13·a_1_0, an element of degree 7
  4. a_7_1 → c_2_23·a_1_0 − c_2_13·a_1_1, an element of degree 7
  5. c_8_0 → c_2_24 − c_2_12·c_2_22 + c_2_14, an element of degree 8
  6. c_8_1 → c_2_1·c_2_23 − c_2_13·c_2_2, an element of degree 8
  7. a_11_0 → c_2_25·a_1_1 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0 + c_2_15·a_1_0, an element of degree 11
  8. c_12_0 → c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12

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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