Mod-19-Cohomology of group number 1 of order 114

About the group Ring generators Ring relations Completion information Restriction maps


General information on the group

  • The group order factors as 2 · 3 · 19.
  • It is non-abelian.
  • It has 19-Rank 1.
  • The centre of a Sylow 19-subgroup has rank 1.
  • Its Sylow 19-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 5 stability conditions for H*(SmallGroup(19,1); GF(19)).

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  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10)

    ( − 1  +  t) · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4)
  • 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].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 2 minimal generators of maximal degree 12:

  1. a_11_0, a nilpotent element of degree 11
  2. c_12_0, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one "obvious" relation:
   a_11_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 12 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_12_0, an element of degree 12
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 11].


About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(19,1); GF(19))

  1. a_11_0c_2_05·a_1_0
  2. c_12_0c_2_06

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

  1. a_11_0c_2_05·a_1_0, an element of degree 11
  2. c_12_0c_2_06, an element of degree 12


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