Cohomology of group number 49 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 81gp6xC3, the Direct product 81gp6 x C_3.
  • The group has 3 minimal generators and exponent 27.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
     − 1
    (t  −  1)3 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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

Ring generators

The cohomology ring has 8 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. a_1_2, a nilpotent element of degree 1
  4. b_2_3, an element of degree 2
  5. c_2_4, a Duflot regular element of degree 2
  6. a_3_6, a nilpotent element of degree 3
  7. a_5_10, a nilpotent element of degree 5
  8. c_6_13, a Duflot regular element of degree 6

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

Ring relations

There are 5 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_62, a_5_102

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

  1. b_2_3·a_1_0
  2. a_1_0·a_3_6
  3. b_2_3·a_3_6
  4. a_1_0·a_5_10
  5. a_3_6·a_5_10

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

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_2_4, a Duflot regular element of degree 2
    2. c_6_13, a Duflot regular element of degree 6
    3. b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → a_1_0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. c_2_4 → c_2_1, an element of degree 2
  6. a_3_6 → 0, an element of degree 3
  7. a_5_10 → 0, an element of degree 5
  8. c_6_13 →  − c_2_23, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → a_1_2, an element of degree 1
  3. a_1_2 → a_1_0, an element of degree 1
  4. b_2_3 → c_2_5, an element of degree 2
  5. c_2_4 → c_2_3, an element of degree 2
  6. a_3_6 → 0, an element of degree 3
  7. a_5_10 → 0, an element of degree 5
  8. c_6_13 → c_2_4·c_2_52 − c_2_43, an element of degree 6

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


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