Mod-3-Cohomology of SymmetricGroup(8), a group of order 40320

About the group Ring generators Ring relations Completion information Restriction maps


General information on the group

  • SymmetricGroup(8) is a group of order 40320.
  • The group order factors as 27 · 32 · 5 · 7.
  • The group is defined by Group([(1,2,3,4,5,6,7,8),(1,2)]).
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 2.
  • Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.


Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(144,186); GF(3)).

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  +  t2  −  t3  +  t4  −  t5  +  t6)

    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)
  • 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 4 minimal generators of maximal degree 8:

  1. a_3_0, a nilpotent element of degree 3
  2. c_4_0, a Duflot element of degree 4
  3. a_7_1, a nilpotent element of degree 7
  4. c_8_1, a Duflot element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 2 "obvious" relations:
   a_3_02, a_7_12

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 10 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 0 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_0, an element of degree 4
    2. c_8_1, an element of degree 8
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].


About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(144,186); GF(3))

  1. a_3_0a_3_0
  2. c_4_0c_4_0
  3. a_7_1a_7_1
  4. c_8_1c_8_1

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

  1. a_3_0c_2_2·a_1_1 + c_2_1·a_1_0, an element of degree 3
  2. c_4_0c_2_22 + c_2_12, an element of degree 4
  3. a_7_1c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1, an element of degree 7
  4. c_8_1c_2_12·c_2_22, an element of degree 8


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