Mod-2-Cohomology of group number 43 of order 168

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General information on the group

  • The group order factors as 23 · 3 · 7.
  • It is non-abelian.
  • It has 2-Rank 3.
  • The centre of a Sylow 2-subgroup has rank 3.
  • Its Sylow 2-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

The computation was based on 20 stability conditions for H*(SmallGroup(8,5); GF(2)).

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·((1  −  t  +  t2  −  t3  +  t4) · (1  −  t2  +  t4))
    ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t  +  t2  +  t3  +  t4  +  t5  +  t6)
  • 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 7:

  1. c_3_0, a Duflot element of degree 3
  2. c_4_0, a Duflot element of degree 4
  3. c_5_0, a Duflot element of degree 5
  4. c_6_1, a Duflot element of degree 6
  5. c_7_1, a Duflot element of degree 7

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Ring relations

There are 2 minimal relations of maximal degree 12:

  1. c_5_02 + c_3_0·c_7_1 + c_4_0·c_6_1 + c_4_0·c_3_02
  2. c_5_0·c_7_1 + c_6_12 + c_6_1·c_3_02 + c_4_03 + c_3_04

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 12 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_3_0, an element of degree 3
    2. c_4_0, an element of degree 4
    3. c_7_1, an element of degree 7
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 11].

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(8,5); GF(2))

  1. c_3_0 → c_1_23 + c_1_12·c_1_2 + c_1_13 + c_1_0·c_1_1·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_2
       + c_1_03
  2. c_4_0 → c_1_24 + c_1_12·c_1_22 + c_1_14 + 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
  3. c_5_0 → c_1_25 + c_1_14·c_1_2 + c_1_15 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_14
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2 + c_1_05
  4. c_6_1 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_24 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22
       + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12
  5. c_7_1 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 3

  1. c_3_0 → c_1_23 + c_1_12·c_1_2 + c_1_13 + c_1_0·c_1_1·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_2
       + c_1_03, an element of degree 3
  2. c_4_0 → c_1_24 + c_1_12·c_1_22 + c_1_14 + 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
  3. c_5_0 → c_1_25 + c_1_14·c_1_2 + c_1_15 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_14
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2 + c_1_05, an element of degree 5
  4. c_6_1 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_24 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22
       + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
  5. c_7_1 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7

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