Mod-3-Cohomology of SymplecticGroup(4,3), a group of order 51840

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General information on the group

  • SymplecticGroup(4,3) is a group of order 51840.
  • The group order factors as 27 · 34 · 5.
  • The group is defined by Group([(1,2)(3,5)(4,7)(6,10)(8,12)(9,13)(11,16)(14,20)(15,21)(17,24)(18,25)(19,27)(23,31)(28,32)(29,37)(30,39)(33,38)(34,43)(35,45)(41,49)(46,54)(47,51)(48,57)(50,53)(52,61)(55,56)(58,67)(59,60)(62,70)(65,71)(66,72)(68,74)(69,75)(73,76)(77,79)(78,80),(1,3,6)(2,4,8)(5,9,14,12,18,26,34,44,25)(7,11,17,10,15,22,30,40,21)(13,19,28,36,47,56,20,29,38)(16,23,32,42,51,60,24,33,37)(27,35,46,55,65,67,73,57,66)(31,41,50,59,68,70,76,61,69)(39,48,58)(43,52,62)(45,53,63)(49,54,64)(71,77,75)(72,74,78)]).
  • 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 1 stability condition for H*(SmallGroup(1296,2895); GF(3)).

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1)·((1  −  t  +  t2  −  t3  +  t4) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6))
    ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t2)2 · (1  +  t4)
  • The a-invariants are -∞,-∞,-∞,-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 6 minimal generators of maximal degree 8:

  1. a_3_0, a nilpotent element of degree 3
  2. b_4_0, an element of degree 4
  3. a_5_0, a nilpotent element of degree 5
  4. c_6_0, a Duflot element of degree 6
  5. a_7_1, a nilpotent element of degree 7
  6. b_8_2, an element of degree 8

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Ring relations

There are 3 "obvious" relations:
   a_3_02, a_5_02, a_7_12

Apart from that, there are no relations.

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 15 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 0 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. b_4_03·b_8_23 − b_4_09 + c_6_02·b_8_23 + b_4_02·c_6_02·b_8_22 + c_6_06, an element of degree 36
    2. b_8_26 + b_4_06·b_8_23 − b_4_0·c_6_02·b_8_24 + b_4_05·c_6_02·b_8_22
         + c_6_04·b_8_23 − b_4_02·c_6_04·b_8_22 + b_4_03·c_6_06, an element of degree 48
    3. c_6_0, an element of degree 6
  • A Duflot regular sequence is given by c_6_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 87].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. b_4_0, an element of degree 4
    2. b_8_2, an element of degree 8
    3. c_6_0, an element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(1296,2895); GF(3))

  1. a_3_0 → a_3_0 − b_2_0·a_1_0
  2. b_4_0 → b_4_0 − b_2_02
  3. a_5_0 → a_5_0 − b_4_0·a_1_0 + b_2_0·a_3_0
  4. c_6_0 → b_2_0·b_4_0 − c_6_4
  5. a_7_1 → b_2_0·a_5_0 + c_6_4·a_1_0
  6. b_8_2 → b_2_0·c_6_4

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_5_0 → 0, an element of degree 5
  4. c_6_0 →  − c_2_03, an element of degree 6
  5. a_7_1 → 0, an element of degree 7
  6. b_8_2 → 0, an element of degree 8

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

  1. a_3_0 →  − c_2_2·a_1_1, an element of degree 3
  2. b_4_0 →  − c_2_22, an element of degree 4
  3. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  4. c_6_0 → c_2_1·c_2_22 − c_2_13, an element of degree 6
  5. a_7_1 → 0, an element of degree 7
  6. b_8_2 → 0, an element of degree 8

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_5_0 →  − c_2_52·a_1_1 + c_2_52·a_1_0 + c_2_4·c_2_5·a_1_2 − c_2_4·c_2_5·a_1_1
       + c_2_4·c_2_5·a_1_0 + c_2_42·a_1_2 − c_2_42·a_1_0 − c_2_3·c_2_5·a_1_2
       + c_2_3·c_2_5·a_1_1 − c_2_3·c_2_5·a_1_0 + c_2_3·c_2_4·a_1_2 + c_2_3·c_2_4·a_1_1
       + c_2_32·a_1_2, an element of degree 5
  4. c_6_0 → c_2_4·c_2_52 − c_2_42·c_2_5 − c_2_3·c_2_52 − c_2_3·c_2_4·c_2_5 + c_2_3·c_2_42
       − c_2_32·c_2_5 − c_2_33, an element of degree 6
  5. a_7_1 → 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
  6. b_8_2 → 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

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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