Mod-5-Cohomology of group number 25 of order 300

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • The group order factors as 22 · 3 · 52.
  • It is non-abelian.
  • It has 5-Rank 2.
  • The centre of a Sylow 5-subgroup has rank 2.
  • Its Sylow 5-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 11 stability conditions for H*(SmallGroup(25,2); GF(5)).

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  +  t2  −  t3  +  t4  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10
    ( − 1  +  t)2 · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4)
  • The a-invariants are -∞,-∞,-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].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 4 minimal generators of maximal degree 12:

  1. a_3_0, a nilpotent element of degree 3
  2. c_4_0, a Duflot element of degree 4
  3. a_11_1, a nilpotent element of degree 11
  4. c_12_1, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 2 "obvious" relations:
   a_3_02, a_11_12

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 14 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 0 and the last generator in degree 12.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_0, an element of degree 4
    2. c_12_1, 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, -1, 14].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(25,2); GF(5))

  1. a_3_0 → c_2_2·a_1_1 − c_2_2·a_1_0 − c_2_1·a_1_1 − c_2_1·a_1_0
  2. c_4_0 → c_2_22 − 2·c_2_1·c_2_2 − c_2_12
  3. a_11_1 → c_2_1·c_2_24·a_1_0 + 2·c_2_12·c_2_23·a_1_1 − c_2_12·c_2_23·a_1_0
       − c_2_13·c_2_22·a_1_1 − 2·c_2_13·c_2_22·a_1_0 − c_2_14·c_2_2·a_1_1
  4. c_12_1 → c_2_12·c_2_24 + c_2_13·c_2_23 − c_2_14·c_2_22

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_3_0 → c_2_2·a_1_1 − c_2_2·a_1_0 − c_2_1·a_1_1 − c_2_1·a_1_0, an element of degree 3
  2. c_4_0 → c_2_22 − 2·c_2_1·c_2_2 − c_2_12, an element of degree 4
  3. a_11_1 → c_2_1·c_2_24·a_1_0 + 2·c_2_12·c_2_23·a_1_1 − c_2_12·c_2_23·a_1_0
       − c_2_13·c_2_22·a_1_1 − 2·c_2_13·c_2_22·a_1_0 − c_2_14·c_2_2·a_1_1, an element of degree 11
  4. c_12_1 → c_2_12·c_2_24 + c_2_13·c_2_23 − c_2_14·c_2_22, 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