Mod-2-Cohomology of U3(3), a group of order 6048

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General information on the group

  • U3(3) is a group of order 6048.
  • The group order factors as 25 · 33 · 7.
  • The group is defined by Group([(2,3)(4,6)(5,8)(7,11)(9,13)(10,15)(12,14)(16,20)(17,22)(18,23)(24,27)(25,28),(1,2,4,7,12,17)(3,5,9,14,19,22)(6,10,13,18,24,23)(8,11,16,21,26,28)(20,25,27)]).
  • 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 1 stability condition for H*(SmallGroup(96,67); 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  −  t3  +  t4

    ( − 1  +  t)2 · (1  +  t2) · (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 4 minimal generators of maximal degree 6:

  1. a_3_0, a nilpotent element of degree 3
  2. c_4_0, a Duflot element of degree 4
  3. a_5_0, a nilpotent element of degree 5
  4. b_6_0, an element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 2 minimal relations of maximal degree 10:

  1. a_3_02
  2. a_5_02


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.
  • 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_4_0, an element of degree 4
    2. b_6_0, an element of degree 6
  • A Duflot regular sequence is given by c_4_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 8].


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Restriction maps

Expressing the generators as elements of H*(SmallGroup(96,67); GF(2))

  1. a_3_0a_3_0
  2. c_4_0b_2_02 + a_1_0·a_3_0 + c_4_2
  3. a_5_0b_2_0·a_3_0 + c_4_2·a_1_0
  4. b_6_0b_2_0·a_1_0·a_3_0 + b_2_0·c_4_2

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

  1. a_3_00, an element of degree 3
  2. c_4_0c_1_04, an element of degree 4
  3. a_5_00, an element of degree 5
  4. b_6_00, an element of degree 6

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

  1. a_3_00, an element of degree 3
  2. c_4_0c_1_14 + c_1_02·c_1_12 + c_1_04, an element of degree 4
  3. a_5_00, an element of degree 5
  4. b_6_0c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6

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

  1. a_3_00, an element of degree 3
  2. c_4_0c_1_14 + c_1_02·c_1_12 + c_1_04, an element of degree 4
  3. a_5_00, an element of degree 5
  4. b_6_0c_1_02·c_1_14 + c_1_04·c_1_12, 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