Mod-2-Cohomology of group number 8282 of order 576

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General information on the group

  • The group order factors as 26 · 32.
  • It is non-abelian.
  • It has 2-Rank 3.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

The computation was based on 4 stability conditions for H*(Syl2(M12); GF(2)).

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)
    ( − 1  +  t)3 · (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 5 minimal generators of maximal degree 4:

  1. b_1_0, an element of degree 1
  2. b_2_0, an element of degree 2
  3. b_3_1, an element of degree 3
  4. b_3_0, an element of degree 3
  5. c_4_4, a Duflot element of degree 4

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Ring relations

There are 2 minimal relations of maximal degree 6:

  1. b_1_0·b_3_1
  2. b_3_0·b_3_1 + b_3_02 + b_1_03·b_3_0 + b_2_0·b_1_0·b_3_0 + c_4_4·b_1_02

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 6 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_4_4, an element of degree 4
    2. b_1_02 + b_2_0, an element of degree 2
    3. b_3_1 + b_2_0·b_1_0, an element of degree 3
  • A Duflot regular sequence is given by c_4_4.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 6].

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Restriction maps

Expressing the generators as elements of H*(Syl2(M12); GF(2))

  1. b_1_0 → b_1_0
  2. b_2_0 → b_1_22 + b_1_1·b_1_2 + b_1_12 + b_2_5
  3. b_3_1 → b_1_1·b_1_22 + b_1_12·b_1_2 + b_2_5·b_1_2
  4. b_3_0 → b_3_9 + b_2_5·b_1_2 + b_2_4·b_1_1
  5. c_4_4 → b_2_4·b_1_1·b_1_2 + b_2_42 + c_4_14

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

  1. b_1_0 → 0, an element of degree 1
  2. b_2_0 → 0, an element of degree 2
  3. b_3_1 → 0, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_4 → c_1_04, an element of degree 4

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

  1. b_1_0 → c_1_1, an element of degree 1
  2. b_2_0 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  3. b_3_1 → 0, an element of degree 3
  4. b_3_0 → c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  5. c_4_4 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_0·c_1_13 + c_1_02·c_1_22
       + c_1_02·c_1_1·c_1_2 + c_1_04, an element of degree 4

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

  1. b_1_0 → 0, an element of degree 1
  2. b_2_0 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
  3. b_3_1 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_4 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + c_1_04, an element of degree 4

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

  1. b_1_0 → 0, an element of degree 1
  2. b_2_0 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
  3. b_3_1 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  4. b_3_0 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  5. c_4_4 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + c_1_04, an element of degree 4

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

  1. b_1_0 → 0, an element of degree 1
  2. b_2_0 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
  3. b_3_1 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  4. b_3_0 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  5. c_4_4 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + c_1_04, an element of degree 4

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

  1. b_1_0 → 0, an element of degree 1
  2. b_2_0 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
  3. b_3_1 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_4 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + c_1_04, an element of degree 4

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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