Mod-3-Cohomology of Sym10, a group of order 3628800

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General information on the group

  • Sym10 is a group of order 3628800.
  • The group order factors as 28 · 34 · 52 · 7.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9,10),(1,2)]).
  • It is non-abelian.
  • It has 3-Rank 3.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

The computation was based on 4 stability conditions for H*(SmallGroup(324,39); GF(3)).

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1)·(1  −  3·t  +  6·t2  −  9·t3  +  12·t4  −  15·t5  +  18·t6  −  20·t7  +  22·t8  −  24·t9  +  28·t10  −  31·t11  +  33·t12  −  34·t13  +  34·t14  −  32·t15  +  29·t16  −  25·t17  +  22·t18  −  20·t19  +  18·t20  −  15·t21  +  12·t22  −  9·t23  +  6·t24  −  3·t25  +  t26)
    ( − 1  +  t)3 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)3 · (1  −  t2  +  t4) · (1  +  t4) · (1  +  t8)
  • The a-invariants are -∞,-∞,-12,-3. 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, -3, -3].

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 16:

  1. a_3_0, a nilpotent element of degree 3
  2. b_4_0, an element of degree 4
  3. a_7_0, a nilpotent element of degree 7
  4. b_8_0, an element of degree 8
  5. a_10_0, a nilpotent element of degree 10
  6. a_11_1, a nilpotent element of degree 11
  7. a_11_0, a nilpotent element of degree 11
  8. c_12_0, a Duflot element of degree 12
  9. a_15_0, a nilpotent element of degree 15
  10. b_16_3, an element of degree 16

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Ring relations

There are 5 "obvious" relations:
   a_3_02, a_7_02, a_11_02, a_11_12, a_15_02

Apart from that, there are 24 minimal relations of maximal degree 27:

  1. a_10_0·a_3_0
  2. a_3_0·a_11_0
  3. b_4_0·a_10_0
  4. b_4_0·a_11_0
  5. a_10_0·a_7_0
  6. a_3_0·a_15_0
  7. a_7_0·a_11_0
  8. b_8_0·a_10_0
  9. b_4_0·a_15_0
  10. b_8_0·a_11_0
  11. b_16_3·a_3_0
  12. a_10_02
  13. b_4_0·b_16_3
  14. a_10_0·a_11_0
  15. a_10_0·a_11_1
  16. a_7_0·a_15_0
  17. a_11_0·a_11_1
  18. b_8_0·a_15_0
  19. b_16_3·a_7_0
  20. b_8_0·b_16_3
  21. a_10_0·a_15_0
  22. a_11_1·a_15_0
  23. a_10_0·b_16_3 + a_11_0·a_15_0
  24. b_16_3·a_11_1

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 34 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 27 and the last generator in degree 16.
  • The following is a filter regular homogeneous system of parameters:
    1.  − b_4_03·b_8_03 + b_4_09 + b_8_03·c_12_0 + b_4_02·b_8_02·c_12_0 + c_12_03, an element of degree 36
    2. b_16_33 − b_8_06 − b_4_06·b_8_03 − b_4_0·b_8_04·c_12_0 + b_4_05·b_8_02·c_12_0
         − b_8_03·c_12_02 + b_4_02·b_8_02·c_12_02 + b_4_03·c_12_03, an element of degree 48
    3. b_8_0, an element of degree 8
  • A Duflot regular sequence is given by c_12_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 72, 89].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. b_4_03 + c_12_0, an element of degree 12
    2. b_16_3 + b_4_0·c_12_0, an element of degree 16
    3. b_8_0, an element of degree 8

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(324,39); GF(3))

  1. a_3_0 → a_3_1 − a_3_0
  2. b_4_0 → b_4_1 − b_4_0
  3. a_7_0 → a_7_6 + a_7_1
  4. b_8_0 → b_8_4
  5. a_10_0 → a_10_6
  6. a_11_1 → a_11_9 − b_4_1·a_7_1 − b_4_0·b_4_1·a_3_0 + b_4_02·a_3_1
  7. a_11_0 → a_11_13
  8. c_12_0 → b_4_23 + b_4_0·b_8_4 + b_4_02·b_4_1 + c_12_7
  9. a_15_0 → b_4_2·a_11_13 + c_12_7·a_3_2
  10. b_16_3 → b_4_2·c_12_7

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

  1. a_3_0 → 0, an element of degree 3
  2. b_4_0 → 0, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. b_8_0 → 0, an element of degree 8
  5. a_10_0 → 0, an element of degree 10
  6. a_11_1 → 0, an element of degree 11
  7. a_11_0 → 0, an element of degree 11
  8. c_12_0 → c_2_06, an element of degree 12
  9. a_15_0 → 0, an element of degree 15
  10. b_16_3 → 0, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0 → 0, an element of degree 3
  2. b_4_0 → 0, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. b_8_0 → 0, an element of degree 8
  5. a_10_0 →  − c_2_1·c_2_23·a_1_0·a_1_1 + c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  6. a_11_1 → 0, an element of degree 11
  7. a_11_0 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, 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
  9. a_15_0 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  10. b_16_3 → c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. a_3_0 → c_2_5·a_1_2 + c_2_5·a_1_1 − c_2_5·a_1_0 + c_2_4·a_1_2 + c_2_4·a_1_1 − c_2_3·a_1_2, an element of degree 3
  2. b_4_0 → c_2_52 − c_2_4·c_2_5 + c_2_42 + c_2_3·c_2_5, an element of degree 4
  3. a_7_0 → c_2_4·c_2_52·a_1_0 − c_2_42·c_2_5·a_1_0 + c_2_3·c_2_52·a_1_1 − c_2_3·c_2_52·a_1_0
       − c_2_3·c_2_4·c_2_5·a_1_2 + c_2_3·c_2_4·c_2_5·a_1_1 − c_2_3·c_2_42·a_1_2
       − c_2_32·c_2_5·a_1_2 + c_2_33·a_1_2, an element of degree 7
  4. b_8_0 → c_2_3·c_2_4·c_2_52 − c_2_3·c_2_42·c_2_5 + c_2_32·c_2_52 + c_2_33·c_2_5, an element of degree 8
  5. a_10_0 → 0, an element of degree 10
  6. a_11_1 → c_2_4·c_2_54·a_1_1 + c_2_4·c_2_54·a_1_0 − c_2_42·c_2_53·a_1_2
       − c_2_42·c_2_53·a_1_0 − c_2_43·c_2_52·a_1_1 + c_2_44·c_2_5·a_1_2
       + c_2_3·c_2_54·a_1_1 − c_2_3·c_2_54·a_1_0 + c_2_3·c_2_4·c_2_53·a_1_2
       + c_2_3·c_2_4·c_2_53·a_1_1 + c_2_3·c_2_42·c_2_52·a_1_0 + c_2_3·c_2_43·c_2_5·a_1_0
       + c_2_3·c_2_44·a_1_0 + c_2_32·c_2_53·a_1_2 + c_2_32·c_2_4·c_2_52·a_1_1
       + c_2_32·c_2_42·c_2_5·a_1_2 − c_2_32·c_2_43·a_1_2 − c_2_32·c_2_43·a_1_1
       + c_2_33·c_2_52·a_1_1 − c_2_33·c_2_52·a_1_0 − c_2_33·c_2_4·c_2_5·a_1_2
       + c_2_33·c_2_4·c_2_5·a_1_1 + c_2_33·c_2_4·c_2_5·a_1_0 − c_2_33·c_2_42·a_1_2
       − c_2_33·c_2_42·a_1_0 + c_2_34·c_2_5·a_1_2 + c_2_34·c_2_5·a_1_1
       − c_2_34·c_2_5·a_1_0 + c_2_34·c_2_4·a_1_2 + c_2_34·c_2_4·a_1_1 + c_2_35·a_1_2, an element of degree 11
  7. a_11_0 → 0, an element of degree 11
  8. c_12_0 → c_2_42·c_2_54 + c_2_43·c_2_53 + c_2_44·c_2_52 + c_2_3·c_2_4·c_2_54
       + c_2_3·c_2_43·c_2_52 + c_2_3·c_2_44·c_2_5 + c_2_32·c_2_54
       + c_2_32·c_2_42·c_2_52 + c_2_32·c_2_43·c_2_5 + c_2_32·c_2_44
       − c_2_33·c_2_53 − c_2_34·c_2_4·c_2_5 + c_2_34·c_2_42 − c_2_35·c_2_5 + c_2_36, an element of degree 12
  9. a_15_0 → 0, an element of degree 15
  10. b_16_3 → 0, an element of degree 16

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