Mod-13-Cohomology of L3(3).2, a group of order 11232

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • L3(3).2 is a group of order 11232.
  • The group order factors as 25 · 33 · 13.
  • The group is defined by Group([(1,2)(3,5)(4,6)(7,10)(8,9)(11,15)(12,16)(13,18)(14,20)(17,22)(19,24)(21,23)(25,26),(1,3)(2,4,7,11)(5,8,12,17)(6,9,13,19)(10,14)(15,21,25,22)(16,20)(18,23)(24,26)]).
  • It is non-abelian.
  • It has 13-Rank 1.
  • The centre of a Sylow 13-subgroup has rank 1.
  • Its Sylow 13-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(78,1); GF(13)).

General information

  • The cohomology ring is of dimension 1 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·(1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10)
    ( − 1  +  t) · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4)
  • The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -1].

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Ring generators

The cohomology ring has 2 minimal generators of maximal degree 12:

  1. a_11_0, a nilpotent element of degree 11
  2. c_12_0, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one "obvious" relation:
   a_11_02

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 12 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_12_0, an element of degree 12
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 11].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(78,1); GF(13))

  1. a_11_0 → a_11_0
  2. c_12_0 → c_12_0

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

  1. a_11_0 → c_2_05·a_1_0, an element of degree 11
  2. c_12_0 → c_2_06, an element of degree 12

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