Cohomology of group number 24 of order 243

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243


General information on the group

  • The group is also known as Mod243, the Modular group of order 243.
  • The group has 2 minimal generators and exponent 81.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.


Structure of the cohomology ring

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  −  1)2 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 6 minimal generators of maximal degree 6:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_2_1, an element of degree 2
  4. a_3_1, a nilpotent element of degree 3
  5. a_5_1, a nilpotent element of degree 5
  6. c_6_2, a Duflot regular element of degree 6

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Ring relations

There are 4 "obvious" relations:
   a_1_02, a_1_12, a_3_12, a_5_12

Apart from that, there are 5 minimal relations of maximal degree 8:

  1. b_2_1·a_1_0
  2. a_1_0·a_3_1
  3. b_2_1·a_3_1
  4. a_1_0·a_5_1
  5. a_3_1·a_5_1


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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_6_2, a Duflot regular element of degree 6
    2. b_2_1, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].


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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. b_2_10, an element of degree 2
  4. a_3_10, an element of degree 3
  5. a_5_10, an element of degree 5
  6. c_6_2 − c_2_03, an element of degree 6

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

  1. a_1_00, an element of degree 1
  2. a_1_1a_1_1, an element of degree 1
  3. b_2_1c_2_2, an element of degree 2
  4. a_3_10, an element of degree 3
  5. a_5_10, an element of degree 5
  6. c_6_2c_2_1·c_2_22 − c_2_13, an element of degree 6


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Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany

E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2



Last change: 25.08.2009