Mod-5-Cohomology of U3(4), a group of order 62400

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General information on the group

  • U3(4) is a group of order 62400.
  • The group order factors as 26 · 3 · 52 · 13.
  • The group is defined by Group([(1,2)(3,5)(4,7)(6,10)(8,12)(9,13)(11,16)(14,20)(15,21)(17,24)(18,25)(19,27)(22,31)(23,32)(26,35)(28,37)(29,38)(30,40)(33,44)(34,42)(36,47)(39,51)(41,43)(45,55)(46,52)(48,57)(49,58)(50,59)(53,62)(54,60)(56,61)(63,64),(1,3,6)(2,4,8)(5,9,14)(7,11,17)(10,15,22)(12,18,26)(13,19,28)(16,23,33)(20,29,39)(21,30,41)(25,34,45)(27,36,48)(31,42,53)(32,43,54)(35,46,56)(37,49,57)(38,50,60)(40,52,47)(51,61,58)(55,63,65)(59,64,62)]).
  • It is non-abelian.
  • It has 5-Rank 2.
  • The centre of a Sylow 5-subgroup has rank 2.
  • Its Sylow 5-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(150,5); GF(5)).

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  −  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. c_6_0, a Duflot element of degree 6

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Ring relations

There are 2 "obvious" relations:
   a_3_02, a_5_02

Apart from that, there are no relations.


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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 8 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 0 and the last generator in degree 6.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_0, an element of degree 4
    2. c_6_0, an element of degree 6
  • The above filter regular HSOP forms a Duflot regular sequence.
  • 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(150,5); GF(5))

  1. a_3_0a_3_0
  2. c_4_0c_4_0
  3. a_5_0a_5_0
  4. c_6_0c_6_0

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 + 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1 + c_2_1·a_1_0, an element of degree 3
  2. c_4_0c_2_22 − c_2_1·c_2_2 + c_2_12, an element of degree 4
  3. a_5_0c_2_22·a_1_1 + 2·c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1 − c_2_1·c_2_2·a_1_0
       + 2·c_2_12·a_1_1 + c_2_12·a_1_0, an element of degree 5
  4. c_6_0c_2_23 + c_2_1·c_2_22 + c_2_12·c_2_2 + c_2_13, an element of degree 6


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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