Mod-2-Cohomology of AlternatingGroup(6), a group of order 360

About the group Ring generators Ring relations Completion information Restriction maps


General information on the group

  • AlternatingGroup(6) is a group of order 360.
  • The group order factors as 23 · 32 · 5.
  • The group is defined by Group([(1,2,3,4,5),(4,5,6)]).
  • It is non-abelian.
  • It has 2-Rank 2.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.


Structure of the cohomology ring

The computation was based on 3 stability conditions for H*(D8; GF(2)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • 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].

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Ring generators

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

  1. c_2_0, a Duflot element of degree 2
  2. b_3_1, an element of degree 3
  3. b_3_0, an element of degree 3

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Ring relations

There is one minimal relation of degree 6:

  1. b_3_0·b_3_1


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. b_3_1 + b_3_0, an element of degree 3
  • A Duflot regular sequence is given by c_2_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 3].


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Restriction maps

Expressing the generators as elements of H*(D8; GF(2))

  1. c_2_0b_1_12 + b_1_02 + c_2_2
  2. b_3_1c_2_2·b_1_0
  3. b_3_0c_2_2·b_1_1

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

  1. c_2_0c_1_02, an element of degree 2
  2. b_3_10, an element of degree 3
  3. b_3_00, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. c_2_0c_1_12 + c_1_0·c_1_1 + c_1_02, an element of degree 2
  2. b_3_1c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  3. b_3_00, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. c_2_0c_1_12 + c_1_0·c_1_1 + c_1_02, an element of degree 2
  2. b_3_10, an element of degree 3
  3. b_3_0c_1_0·c_1_12 + c_1_02·c_1_1, 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