Mod-3-Cohomology of Normalizer(J2,Centre(SylowSubgroup(J2,3))), a group of order 2160

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General information on the group

  • Normalizer(J2,Centre(SylowSubgroup(J2,3))) is a group of order 2160.
  • The group order factors as 24 · 33 · 5.
  • The group is defined by Group([(3,32,42)(4,86,40)(5,17,51)(6,19,21)(7,65,33)(8,62,72)(9,43,73)(10,59,74)(11,27,66)(12,81,52)(14,18,48)(16,98,96)(20,70,31)(22,56,69)(23,44,67)(24,77,91)(26,90,58)(28,83,46)(29,30,94)(34,53,37)(35,89,76)(36,75,57)(38,64,50)(39,85,84)(45,55,60)(47,49,92)(54,88,80)(61,93,79)(63,78,87)(97,100,99),(2,82,71,15)(4,98,48,87)(5,53)(6,55)(7,44,26,77)(8,22,66,12)(9,30,46,70)(10,36,97,38)(11,81,62,56)(13,41,25,95)(14,63,86,96)(16,18,78,40)(17,37)(19,60)(20,73,29,83)(21,45)(23,58,24,33)(27,52,72,69)(28,31,43,94)(34,51)(39,61,49,54)(47,80,84,79)(50,74,57,99)(59,75,100,64)(65,67,90,91)(85,93,92,88),(1,13,95)(4,8,57)(5,53,89)(6,23,58)(7,60,77)(9,79,83)(10,56,11)(12,98,50)(14,63,97)(15,82,71)(16,64,52)(17,37,76)(18,78,100)(19,44,26)(20,31,70)(21,67,90)(22,66,74)(24,33,55)(25,41,68)(27,59,69)(28,73,93)(29,47,85)(30,49,84)(34,35,51)(36,86,62)(38,81,96)(39,94,92)(40,72,75)(43,61,46)(45,91,65)(48,87,99)(54,88,80),(2,15,68,41,82,95,13,71)(3,4,83,14,65,10,94,57)(5,84,69,80,11,45,12,19)(6,17,85,22,88,27,60,81)(7,59,30,36,42,86,28,18)(8,92,34,79)(9,35,67,64,24,87,31,97)(16,90)(20,99,43,76,23,38,77,78)(21,51,39,56,54,66,55,52)(26,98)(29,75,32,40,46,48,33,74)(37,61,72,47)(44,50,91,63,70,100,73,89)(49,53,93,62)(58,96)]).
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.


Structure of the cohomology ring

The computation was based on 8 stability conditions for H*(E27; GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  −  2·t  +  2·t2  −  t3  +  2·t4  −  3·t5  +  2·t6  −  t7  +  2·t8  −  2·t9  +  t10

    ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  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].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

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

  1. a_3_0, a nilpotent element of degree 3
  2. a_4_1, a nilpotent element of degree 4
  3. b_4_0, an element of degree 4
  4. a_5_0, a nilpotent element of degree 5
  5. a_9_1, a nilpotent element of degree 9
  6. b_10_0, an element of degree 10
  7. a_11_1, a nilpotent element of degree 11
  8. c_12_1, a Duflot element of degree 12

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Ring relations

There are 4 "obvious" relations:
   a_3_02, a_5_02, a_9_12, a_11_12

Apart from that, there are 16 minimal relations of maximal degree 21:

  1. a_4_1·a_3_0
  2. a_4_12
  3. a_4_1·b_4_0 − a_3_0·a_5_0
  4. a_4_1·a_5_0
  5. a_3_0·a_9_1
  6. a_4_1·a_9_1
  7. b_10_0·a_3_0 − b_4_0·a_9_1
  8. a_5_0·a_9_1 − a_3_0·a_11_1
  9. a_4_1·b_10_0 + a_3_0·a_11_1
  10. a_4_1·a_11_1
  11. b_10_0·a_5_0 + b_4_0·a_11_1
  12. a_5_0·a_11_1
  13. b_10_0·a_9_1 + b_4_0·c_12_1·a_3_0
  14. a_9_1·a_11_1 − c_12_1·a_3_0·a_5_0
  15. b_10_02 + b_4_02·c_12_1
  16. b_10_0·a_11_1 − b_4_0·c_12_1·a_5_0


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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 21 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_12_1, an element of degree 12
    2. b_4_0, an element of degree 4
  • A Duflot regular sequence is given by c_12_1.
  • 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*(E27; GF(3))

  1. a_3_0b_2_3·a_1_1 − b_2_1·a_1_1 + b_2_0·a_1_0
  2. a_4_1a_1_1·a_3_5 − a_1_0·a_3_4
  3. b_4_0b_2_32 − b_2_0·b_2_2 − b_2_0·b_2_1 + b_2_02 + a_1_0·a_3_5 − a_1_0·a_3_4
  4. a_5_0b_2_3·a_3_5 − b_2_0·a_3_4
  5. a_9_1b_2_03·b_2_1·a_1_1 − b_2_04·a_1_1 + b_2_3·c_6_8·a_1_1 + b_2_0·c_6_8·a_1_1
       − b_2_0·c_6_8·a_1_0
  6. b_10_0b_2_04·b_2_2 − b_2_03·a_1_1·a_3_5 − b_2_03·a_1_0·a_3_5 − b_2_03·a_1_0·a_3_4
       + b_2_32·c_6_8 + b_2_0·b_2_1·c_6_8 − b_2_02·c_6_8 + c_6_8·a_1_0·a_3_4
  7. a_11_1b_2_03·b_2_1·a_3_5 + b_2_04·a_3_5 − b_2_04·a_3_4 − b_2_04·b_2_1·a_1_1
       + b_2_05·a_1_1 − b_2_3·c_6_8·a_3_5 + b_2_1·c_6_8·a_3_5 − b_2_0·c_6_8·a_3_4
       − b_2_0·b_2_1·c_6_8·a_1_1 + b_2_02·c_6_8·a_1_1
  8. c_12_1b_2_05·b_2_2 − b_2_04·a_1_1·a_3_5 + b_2_04·a_1_0·a_3_5 + b_2_04·a_1_0·a_3_4
       − b_2_02·b_2_2·c_6_8 + b_2_0·c_6_8·a_1_1·a_3_5 + b_2_0·c_6_8·a_1_0·a_3_5
       + b_2_0·c_6_8·a_1_0·a_3_4 − c_6_82

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

  1. a_3_00, an element of degree 3
  2. a_4_10, an element of degree 4
  3. b_4_00, an element of degree 4
  4. a_5_00, an element of degree 5
  5. a_9_10, an element of degree 9
  6. b_10_00, an element of degree 10
  7. a_11_10, an element of degree 11
  8. c_12_1 − c_2_06, an element of degree 12

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

  1. a_3_0c_2_2·a_1_1, an element of degree 3
  2. a_4_1 − c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0c_2_22, an element of degree 4
  4. a_5_0c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1 − c_2_1·c_2_23·a_1_1 + c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0 − c_2_1·c_2_24 + c_2_13·c_2_22, an element of degree 10
  7. a_11_1c_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_1 − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12

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

  1. a_3_0c_2_2·a_1_1, an element of degree 3
  2. a_4_1c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0c_2_22, an element of degree 4
  4. a_5_0 − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1c_2_1·c_2_23·a_1_1 − c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0c_2_1·c_2_24 − c_2_13·c_2_22, an element of degree 10
  7. a_11_1c_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_1 − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12

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

  1. a_3_0c_2_2·a_1_1, an element of degree 3
  2. a_4_1c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0c_2_22, an element of degree 4
  4. a_5_0 − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1c_2_1·c_2_23·a_1_1 − c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0c_2_1·c_2_24 − c_2_13·c_2_22, an element of degree 10
  7. a_11_1c_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_1 − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12

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

  1. a_3_0c_2_2·a_1_1, an element of degree 3
  2. a_4_1 − c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0c_2_22, an element of degree 4
  4. a_5_0c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1 − c_2_1·c_2_23·a_1_1 + c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0 − c_2_1·c_2_24 + c_2_13·c_2_22, an element of degree 10
  7. a_11_1c_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_1 − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12


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