Mod-3-Cohomology of group number 6 of order 54

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General information on the group

  • The group order factors as 2 · 33.
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

The computation was based on 1 stability condition for H*(M27; GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1  −  t  +  t2  −  t5  +  t6  −  t9  +  t10
    ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4)
  • The a-invariants are -∞,-4,-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 6 minimal generators of maximal degree 12:

  1. a_1_0, a nilpotent element of degree 1
  2. b_2_0, an element of degree 2
  3. a_3_0, a nilpotent element of degree 3
  4. a_7_1, a nilpotent element of degree 7
  5. a_11_1, a nilpotent element of degree 11
  6. c_12_2, a Duflot element of degree 12

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Ring relations

There are 4 "obvious" relations:
   a_1_02, a_3_02, a_7_12, a_11_12

Apart from that, there are 5 minimal relations of maximal degree 18:

  1. b_2_0·a_3_0
  2. b_2_0·a_7_1
  3. a_3_0·a_7_1
  4. a_3_0·a_11_1
  5. a_7_1·a_11_1

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 18 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_2, an element of degree 12
    2. b_2_0, an element of degree 2
  • A Duflot regular sequence is given by c_12_2.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 8, 12].

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Restriction maps

Expressing the generators as elements of H*(M27; GF(3))

  1. a_1_0 → a_1_1 − a_1_0
  2. b_2_0 → b_2_1 + a_1_0·a_1_1
  3. a_3_0 → a_3_1
  4. a_7_1 → c_6_2·a_1_0
  5. a_11_1 → c_6_2·a_5_1
  6. c_12_2 → c_6_22

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

  1. a_1_0 → 0, an element of degree 1
  2. b_2_0 → 0, an element of degree 2
  3. a_3_0 → 0, an element of degree 3
  4. a_7_1 → 0, an element of degree 7
  5. a_11_1 → 0, an element of degree 11
  6. c_12_2 → c_2_06, an element of degree 12

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

  1. a_1_0 → a_1_1, an element of degree 1
  2. b_2_0 → c_2_2, an element of degree 2
  3. a_3_0 → 0, an element of degree 3
  4. a_7_1 → 0, an element of degree 7
  5. a_11_1 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  6. c_12_2 →  − c_2_1·c_2_24·a_1_0·a_1_1 + c_2_13·c_2_22·a_1_0·a_1_1 + c_2_12·c_2_24
       + c_2_14·c_2_22 + c_2_16, an element of degree 12

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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