Mod-2-Cohomology of group number 5602 of order 384

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General information on the group

  • The group order factors as 27 · 3.
  • It is non-abelian.
  • It has 2-Rank 4.
  • 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 of rank 3, 3, 4, 4 and 4, respectively.

Structure of the cohomology ring

The computation was based on 1 stability condition for H*(Syl2(S8); GF(2)).

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  +  2·t2  +  t3  +  t5
    (1  +  t) · ( − 1  +  t)4 · (1  +  t2) · (1  +  t  +  t2)
  • The a-invariants are -∞,-∞,-∞,-6,-4. 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, -4, -4].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 4:

  1. b_1_1, an element of degree 1
  2. b_1_0, an element of degree 1
  3. b_2_4, an element of degree 2
  4. b_2_3, an element of degree 2
  5. b_3_9, an element of degree 3
  6. b_3_1, an element of degree 3
  7. b_3_0, an element of degree 3
  8. c_4_15, a Duflot element of degree 4

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Ring relations

There are 10 minimal relations of maximal degree 6:

  1. b_2_3·b_1_1
  2. b_1_0·b_3_9
  3. b_1_1·b_3_0 + b_1_0·b_3_1
  4. b_1_1·b_3_9
  5. b_2_3·b_3_1
  6. b_2_4·b_3_9
  7. b_3_02 + b_2_4·b_1_0·b_3_0 + b_2_3·b_2_42 + c_4_15·b_1_02
  8. b_3_0·b_3_1 + b_2_4·b_1_0·b_3_1 + c_4_15·b_1_0·b_1_1
  9. b_3_0·b_3_9
  10. b_3_1·b_3_9

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 9 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 6 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_15, an element of degree 4
    2. b_1_1·b_3_1 + b_1_14 + b_1_0·b_3_0 + b_1_02·b_1_12 + b_1_04 + b_2_4·b_1_0·b_1_1
         + b_2_4·b_1_02 + b_2_42 + b_2_32, an element of degree 4
    3. b_3_92 + b_3_12 + b_1_13·b_3_1 + b_1_0·b_1_12·b_3_1 + b_1_02·b_1_14
         + b_1_03·b_3_1 + b_1_03·b_3_0 + b_1_04·b_1_12 + b_2_4·b_1_1·b_3_1
         + b_2_4·b_1_0·b_1_13 + b_2_4·b_1_04 + b_2_42·b_1_12 + b_2_42·b_1_0·b_1_1
         + b_2_42·b_1_02 + b_2_3·b_1_0·b_3_0 + b_2_3·b_2_4·b_1_02 + b_2_3·b_2_42
         + b_2_32·b_1_02 + c_4_15·b_1_12, an element of degree 6
    4. b_1_1 + b_1_0, an element of degree 1
  • A Duflot regular sequence is given by c_4_15.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 8, 11].
  • We found that there exists some HSOP over a finite extension field, in degrees 4,4,1,3.

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Restriction maps

Expressing the generators as elements of H*(Syl2(S8); GF(2))

  1. b_1_1 → b_1_1
  2. b_1_0 → b_1_2
  3. b_2_4 → b_2_6 + b_2_4
  4. b_2_3 → b_1_02 + b_2_5
  5. b_3_9 → b_2_5·b_1_0
  6. b_3_1 → b_3_11
  7. b_3_0 → b_3_12 + b_2_4·b_1_0
  8. c_4_15 → c_4_21

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

  1. b_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_4 → 0, an element of degree 2
  4. b_2_3 → 0, an element of degree 2
  5. b_3_9 → 0, an element of degree 3
  6. b_3_1 → 0, an element of degree 3
  7. b_3_0 → 0, an element of degree 3
  8. c_4_15 → 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_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_4 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  4. b_2_3 → c_1_22, an element of degree 2
  5. b_3_9 → 0, an element of degree 3
  6. b_3_1 → 0, an element of degree 3
  7. b_3_0 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. c_4_15 → 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_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_4 → 0, an element of degree 2
  4. b_2_3 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
  5. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  6. b_3_1 → 0, an element of degree 3
  7. b_3_0 → 0, an element of degree 3
  8. c_4_15 → 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 4 in a Sylow subgroup

  1. b_1_1 → c_1_2, an element of degree 1
  2. b_1_0 → c_1_3, an element of degree 1
  3. b_2_4 → c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_3 + c_1_0·c_1_2, an element of degree 2
  4. b_2_3 → 0, an element of degree 2
  5. b_3_9 → 0, an element of degree 3
  6. b_3_1 → c_1_1·c_1_2·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_2·c_1_3 + c_1_02·c_1_2, an element of degree 3
  7. b_3_0 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  8. c_4_15 → c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_1·c_1_3 + c_1_02·c_1_12 + c_1_03·c_1_3 + c_1_03·c_1_2 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_1 → c_1_2, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_4 → c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_2, an element of degree 2
  4. b_2_3 → 0, an element of degree 2
  5. b_3_9 → 0, an element of degree 3
  6. b_3_1 → c_1_1·c_1_32 + c_1_1·c_1_2·c_1_3 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_02·c_1_2, an element of degree 3
  7. b_3_0 → 0, an element of degree 3
  8. c_4_15 → c_1_0·c_1_1·c_1_32 + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_3
       + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_3
       + c_1_02·c_1_12 + c_1_03·c_1_2 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_1 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_2_4 → c_1_1·c_1_3 + c_1_1·c_1_2 + c_1_12 + c_1_0·c_1_2, an element of degree 2
  4. b_2_3 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  5. b_3_9 → 0, an element of degree 3
  6. b_3_1 → 0, an element of degree 3
  7. b_3_0 → c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_22
       + c_1_02·c_1_2, an element of degree 3
  8. c_4_15 → c_1_0·c_1_1·c_1_32 + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_3
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_3 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + c_1_03·c_1_2 + c_1_04, an element of degree 4

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