Mod-2-Cohomology of group number 3 of order 12

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • The group order factors as 22 · 3.
  • It is non-abelian.
  • It has 2-Rank 2.
  • The centre of a Sylow 2-subgroup has rank 2.
  • Its Sylow 2-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

The computation was based on 2 stability conditions for H*(SmallGroup(4,2); GF(2)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1  −  t  +  t2
    ( − 1  +  t)2 · (1  +  t  +  t2)
  • The a-invariants are -∞,-∞,-2. 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, -2].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

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

  1. c_2_0, a Duflot element of degree 2
  2. c_3_1, a Duflot element of degree 3
  3. c_3_0, a Duflot element of degree 3

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one minimal relation of degree 6:

  1. c_3_12 + c_3_0·c_3_1 + c_3_02 + c_2_03

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 6 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_2_0, an element of degree 2
    2. c_3_1, an element of degree 3
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 3].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(4,2); GF(2))

  1. c_2_0 → c_1_12 + c_1_0·c_1_1 + c_1_02
  2. c_3_1 → c_1_0·c_1_12 + c_1_02·c_1_1
  3. c_3_0 → c_1_13 + c_1_02·c_1_1 + c_1_03

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. c_2_0 → c_1_12 + c_1_0·c_1_1 + c_1_02, an element of degree 2
  2. c_3_1 → c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  3. c_3_0 → c_1_13 + c_1_02·c_1_1 + c_1_03, an element of degree 3

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