Cohomology of group number 13 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 is also known as M125xC5, the Direct product M125 x C_5.
  • The group has 3 minimal generators and exponent 25.
  • 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 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)

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

Ring generators

The cohomology ring has 10 minimal generators of maximal degree 10:

  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. a_7_14, a nilpotent element of degree 7
  9. a_9_18, a nilpotent element of degree 9
  10. c_10_21, a Duflot regular element of degree 10

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

Ring relations

There are 7 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_62, a_5_102, a_7_142, a_9_182

Apart from that, there are 14 minimal relations of maximal degree 16:

  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. b_2_3·a_5_10
  6. a_3_6·a_5_10
  7. a_1_0·a_7_14
  8. b_2_3·a_7_14
  9. a_3_6·a_7_14
  10. a_1_0·a_9_18
  11. a_5_10·a_7_14
  12. a_3_6·a_9_18
  13. a_5_10·a_9_18
  14. a_7_14·a_9_18

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 16.
  • 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_10_21, a Duflot regular element of degree 10
    3. b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • 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 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. 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. a_7_14 → 0, an element of degree 7
  9. a_9_18 → 0, an element of degree 9
  10. c_10_21 →  − c_2_25, an element of degree 10

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. a_7_14 → 0, an element of degree 7
  9. a_9_18 → c_2_54·a_1_1 − c_2_4·c_2_53·a_1_2, an element of degree 9
  10. c_10_21 →  − c_2_54·a_1_1·a_1_2 + c_2_4·c_2_54 − c_2_45, an element of degree 10

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