Cohomology of group number 47 of order 32

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 32

General information on the group

  • The group is also known as Q8xV4, the Direct product Q8 x V_4.
  • The group has 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 3.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t2  +  t  +  1)
    (t  −  1)3 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-3. 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 32

Ring generators

The cohomology ring has 5 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. c_1_2, a Duflot regular element of degree 1
  4. c_1_3, a Duflot regular element of degree 1
  5. c_4_21, a Duflot regular element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 32

Ring relations

There are 2 minimal relations of maximal degree 3:

  1. a_1_12 + a_1_0·a_1_1 + a_1_02
  2. a_1_03

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 32

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 5.
  • However, the last relation was already found in degree 3 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_1_2, a Duflot regular element of degree 1
    2. c_1_3, a Duflot regular element of degree 1
    3. c_4_21, a Duflot regular element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 32

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. c_1_3 → c_1_1, an element of degree 1
  5. c_4_21 → c_1_24, an element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 32


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