Cohomology of group number 3 of order 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 8

General information on the group

  • The group is also known as D8, the Dihedral group of order 8.
  • The group has 2 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

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
    (t  −  1)2
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 8

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 regular element of degree 2

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 8

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 Back to groups of order 8

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 3.
  • 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, a Duflot regular element of degree 2
    2. b_1_1 + b_1_0, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 1].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 8

Restriction maps

Restriction map to the greatest central el. ab. subgp., 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

  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

  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 Back to groups of order 8


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009