Mod-3-Cohomology of group number 2895 of order 1296

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General information on the group

  • The group order factors as 24 · 34.
  • 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*(Syl3(A9); 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)3 · (1  +  t2) · (1  +  t  +  t2)
  • 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 6:

  1. a_1_0, a nilpotent element of degree 1
  2. b_2_0, an element of degree 2
  3. a_3_0, a nilpotent element of degree 3
  4. b_4_0, an element of degree 4
  5. a_5_0, a nilpotent element of degree 5
  6. c_6_4, a Duflot element of degree 6

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Ring relations

There are 3 "obvious" relations:
   a_1_02, a_3_02, a_5_02

Apart from that, there are no relations.

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 10 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 0 and the last generator in degree 6.
  • The following is a filter regular homogeneous system of parameters:
    1.  − b_4_09 + b_2_06·b_4_06 + b_2_018 + b_2_04·b_4_04·c_6_42
         + b_2_06·b_4_03·c_6_42 + b_2_08·b_4_02·c_6_42 − b_2_05·b_4_02·c_6_43
         + b_2_07·b_4_0·c_6_43 − b_2_09·c_6_43 + b_2_02·b_4_02·c_6_44
         − b_2_04·b_4_0·c_6_44 + b_2_06·c_6_44 + b_2_03·c_6_45 + c_6_46, an element of degree 36
    2. b_2_06·b_4_09 − b_2_012·b_4_06 + b_2_04·b_4_07·c_6_42
         + b_2_010·b_4_04·c_6_42 − b_2_012·b_4_03·c_6_42 − b_2_014·b_4_02·c_6_42
         − b_2_05·b_4_05·c_6_43 − b_2_011·b_4_02·c_6_43 − b_2_013·b_4_0·c_6_43
         + b_2_015·c_6_43 + b_2_02·b_4_05·c_6_44 + b_2_04·b_4_04·c_6_44
         − b_2_06·b_4_03·c_6_44 − b_2_010·b_4_0·c_6_44 − b_2_012·c_6_44
         + b_2_03·b_4_03·c_6_45 − b_2_07·b_4_0·c_6_45 + b_4_03·c_6_46
         − b_2_02·b_4_02·c_6_46 + b_2_03·c_6_47, an element of degree 48
    3. b_2_0, an element of degree 2
  • A Duflot regular sequence is given by c_6_4.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 83].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1.  − b_4_09 + b_2_06·b_4_06 + b_2_018 + b_2_04·b_4_04·c_6_42
         + b_2_06·b_4_03·c_6_42 + b_2_08·b_4_02·c_6_42 − b_2_05·b_4_02·c_6_43
         + b_2_07·b_4_0·c_6_43 − b_2_09·c_6_43 + b_2_02·b_4_02·c_6_44
         − b_2_04·b_4_0·c_6_44 + b_2_06·c_6_44 + b_2_03·c_6_45 + c_6_46, an element of degree 36
    2. b_4_0, an element of degree 4
    3. b_2_0, an element of degree 2
  • We found that there exists some HSOP over a finite extension field, in degrees 4,2,6.

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Restriction maps

Expressing the generators as elements of H*(Syl3(A9); GF(3))

  1. a_1_0 → a_1_1
  2. b_2_0 → b_2_2
  3. a_3_0 → a_3_4 − b_2_0·a_1_0
  4. b_4_0 → b_4_6 − b_2_02
  5. a_5_0 → a_5_8 + b_2_0·a_3_3
  6. c_6_4 → c_6_11

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

  1. a_1_0 → 0, an element of degree 1
  2. b_2_0 → 0, an element of degree 2
  3. a_3_0 → 0, an element of degree 3
  4. b_4_0 → 0, an element of degree 4
  5. a_5_0 → 0, an element of degree 5
  6. c_6_4 → c_2_03, an element of degree 6

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

  1. a_1_0 → 0, an element of degree 1
  2. b_2_0 → 0, an element of degree 2
  3. a_3_0 →  − c_2_2·a_1_1, an element of degree 3
  4. b_4_0 →  − c_2_22, an element of degree 4
  5. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  6. c_6_4 →  − c_2_1·c_2_22 + c_2_13, an element of degree 6

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

  1. a_1_0 → a_1_2, an element of degree 1
  2. b_2_0 → c_2_5, an element of degree 2
  3. a_3_0 →  − 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
  4. b_4_0 → c_2_4·c_2_5 − c_2_42 − c_2_3·c_2_5, an element of degree 4
  5. a_5_0 → c_2_4·c_2_5·a_1_0 − c_2_42·a_1_0 + 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
  6. c_6_4 → 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

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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