Mod-7-Cohomology of group number 2 of order 42

About the group Ring generators Ring relations Completion information Restriction maps


General information on the group

  • The group order factors as 2 · 3 · 7.
  • It is non-abelian.
  • It has 7-Rank 1.
  • The centre of a Sylow 7-subgroup has rank 1.
  • Its Sylow 7-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.


Structure of the cohomology ring

The computation was based on 2 stability conditions for H*(SmallGroup(7,1); GF(7)).

General information

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

    ( − 1  +  t) · (1  −  t  +  t2) · (1  +  t  +  t2)
  • The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -1].

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Ring generators

The cohomology ring has 2 minimal generators of maximal degree 6:

  1. a_5_0, a nilpotent element of degree 5
  2. c_6_0, a Duflot element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one "obvious" relation:
   a_5_02

Apart from that, there are no relations.


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_6_0, an element of degree 6
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 5].


About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(7,1); GF(7))

  1. a_5_0c_2_02·a_1_0
  2. c_6_0c_2_03

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_5_0c_2_02·a_1_0, an element of degree 5
  2. c_6_0c_2_03, an element of degree 6


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