Mod-2-Cohomology of group number 8 of order 24

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • The group order factors as 23 · 3.
  • It is non-abelian.
  • It has 2-Rank 2.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(D8; GF(2)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1
    ( − 1  +  t)2
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 3 minimal generators of maximal degree 2:

  1. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. c_2_2, a Duflot element of degree 2

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one minimal relation of degree 2:

  1. b_1_0·b_1_1

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Benson test

  • We proved completion in degree 3 using the Benson criterion.
  • However, the last relation was already found in degree 2 and the last generator in degree 2.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_2, an element of degree 2
    2. b_1_1 + b_1_0, an element of degree 1
  • A Duflot regular sequence is given by c_2_2.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 1].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(D8; GF(2))

  1. b_1_0 → b_1_0
  2. b_1_1 → b_1_1
  3. c_2_2 → c_2_2

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_1_1 → 0, an element of degree 1
  3. c_2_2 → c_1_02, an element of degree 2

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

  1. b_1_0 → c_1_1, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. c_2_2 → c_1_0·c_1_1 + c_1_02, an element of degree 2

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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. c_2_2 → c_1_0·c_1_1 + c_1_02, an element of degree 2

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