Cohomology of group number 4 of order 625

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

General information on the group

  • The group has 2 minimal generators and exponent 25.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 2.
  • 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1
    (t  −  1)2

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

Ring generators

The cohomology ring has 4 minimal generators of maximal degree 2:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. c_2_1, a Duflot regular element of degree 2
  4. c_2_2, a Duflot regular element of degree 2

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

Ring relations

There are 2 "obvious" relations:
   a_1_02, a_1_12

Apart from that, there are no relations.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 2.
  • 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_1, a Duflot regular element of degree 2
    2. c_2_2, a Duflot regular element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 2].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

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. c_2_1 → c_2_2, an element of degree 2
  4. c_2_2 → c_2_1, an element of degree 2

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625


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