Mod-2-Cohomology of SymmetricGroup(7), a group of order 5040

About the group Ring generators Ring relations Completion information Restriction maps


General information on the group

  • SymmetricGroup(7) is a group of order 5040.
  • The group order factors as 24 · 32 · 5 · 7.
  • The group is defined by Group([(1,2,3,4,5,6,7),(1,2)]).
  • It is non-abelian.
  • It has 2-Rank 3.
  • The centre of a Sylow 2-subgroup has rank 2.
  • Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.


Structure of the cohomology ring

The computation was based on 9 stability conditions for H*(D8xC2; GF(2)).

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1)·(1  −  t  +  t2)

    ( − 1  +  t)3 · (1  +  t  +  t2)
  • The a-invariants are -∞,-∞,-∞,-3. 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, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 4 minimal generators of maximal degree 3:

  1. c_1_0, a Duflot element of degree 1
  2. c_2_0, a Duflot element of degree 2
  3. b_3_3, an element of degree 3
  4. c_3_2, 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. b_3_32 + b_3_3·c_3_2 + c_2_0·c_1_0·b_3_3


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_1_0, an element of degree 1
    2. c_2_0, an element of degree 2
    3. c_3_2, an element of degree 3
  • A Duflot regular sequence is given by c_1_0, c_2_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 3].


About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(D8xC2; GF(2))

  1. c_1_0b_1_0 + c_1_2
  2. c_2_0b_1_12 + b_1_1·c_1_2 + c_2_5 + c_1_22
  3. b_3_3c_2_5·b_1_0
  4. c_3_2c_2_5·b_1_1 + b_1_0·c_1_22 + c_2_5·c_1_2 + c_1_23

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

  1. c_1_0c_1_0, an element of degree 1
  2. c_2_0c_1_12 + c_1_02, an element of degree 2
  3. b_3_30, an element of degree 3
  4. c_3_2c_1_0·c_1_12 + c_1_03, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. c_1_0c_1_2 + c_1_0, an element of degree 1
  2. c_2_0c_1_1·c_1_2 + c_1_12 + c_1_02, an element of degree 2
  3. b_3_3c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  4. c_3_2c_1_0·c_1_1·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_2 + c_1_03, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. c_1_0c_1_0, an element of degree 1
  2. c_2_0c_1_22 + c_1_1·c_1_2 + c_1_12 + c_1_0·c_1_2 + c_1_02, an element of degree 2
  3. b_3_30, an element of degree 3
  4. c_3_2c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_1·c_1_2 + c_1_0·c_1_12 + 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