Cohomology of group number 3 of order 125

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General information on the group

  • The group is also known as E125, the Extraspecial 5-group of order 125 and exponent 5.
  • The group has 2 minimal generators and exponent 5.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has 6 conjugacy classes of maximal elementary abelian subgroups, which are all 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
    (t2  +  1) · (t6  +  t2  +  1)
    (t  −  1)2 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)
  • The a-invariants are -∞,-4,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 12 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_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. a_3_4, a nilpotent element of degree 3
  8. a_3_5, a nilpotent element of degree 3
  9. a_7_8, a nilpotent element of degree 7
  10. b_8_9, an element of degree 8
  11. a_9_11, a nilpotent element of degree 9
  12. c_10_12, a Duflot regular element of degree 10

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Ring relations

There are 6 "obvious" relations:
   a_1_02, a_1_12, a_3_42, a_3_52, a_7_82, a_9_112

Apart from that, there are 44 minimal relations of maximal degree 17:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_0
  3. a_2_1·a_1_1
  4. a_2_1·a_1_0 − a_2_0·a_1_1
  5. b_2_3·a_1_0 − b_2_2·a_1_1
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9.  − 2·a_2_1·b_2_2 − a_2_0·b_2_3 + a_1_1·a_3_4
  10. a_2_0·b_2_2 + a_1_0·a_3_4
  11.  − a_2_1·b_2_3 + a_1_1·a_3_5
  12. a_2_1·b_2_2 + 2·a_2_0·b_2_3 + a_1_0·a_3_5
  13. a_2_0·a_3_4
  14.  − b_2_3·a_3_4 + b_2_2·a_3_5 + 2·b_2_2·b_2_3·a_1_1 + 2·b_2_22·a_1_1
  15.  − a_2_1·a_3_4 + a_2_0·a_3_5
  16. a_2_1·a_3_5
  17. a_1_1·a_7_8 + 2·b_2_2·b_2_3·a_1_1·a_3_5 − 2·b_2_22·a_1_1·a_3_5
       + 2·b_2_22·a_1_0·a_3_5
  18. a_1_0·a_7_8 + 2·b_2_22·a_1_1·a_3_5 − 2·b_2_22·a_1_0·a_3_5 + 2·b_2_22·a_1_0·a_3_4
  19. b_2_2·a_7_8 + 2·b_2_2·b_2_33·a_1_1 + 2·b_2_22·b_2_3·a_3_5 + b_2_22·b_2_32·a_1_1
       − 2·b_2_23·a_3_5 + 2·b_2_23·a_3_4 + 2·b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1
  20. a_2_0·a_7_8
  21. a_2_1·a_7_8
  22. b_2_3·a_7_8 + 2·b_2_2·b_2_32·a_3_5 + b_2_2·b_2_33·a_1_1 − 2·b_2_22·b_2_3·a_3_5
       + 2·b_2_22·b_2_32·a_1_1 + 2·b_2_23·a_3_5 − 2·b_2_23·b_2_3·a_1_1 + b_2_24·a_1_1
  23. b_8_9·a_1_1 + b_2_2·b_2_33·a_1_1 − 2·b_2_22·b_2_32·a_1_1 − b_2_24·a_1_1
  24. b_8_9·a_1_0 − b_2_2·b_2_33·a_1_1 + b_2_22·b_2_32·a_1_1 − 2·b_2_23·b_2_3·a_1_1
  25. a_3_5·a_7_8 − b_2_2·b_2_32·a_1_1·a_3_5 − 2·b_2_22·b_2_3·a_1_1·a_3_5
       + 2·b_2_23·a_1_1·a_3_5 − b_2_23·a_1_0·a_3_5
  26. a_3_4·a_7_8 − 2·b_2_2·b_2_32·a_1_1·a_3_5 − 2·b_2_22·b_2_3·a_1_1·a_3_5
       − 2·b_2_23·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_5
  27. b_2_2·b_8_9 − b_2_2·b_2_34 + b_2_22·b_2_33 − 2·b_2_23·b_2_32
       + b_2_22·b_2_3·a_1_1·a_3_5 + b_2_23·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_4
  28. a_2_0·b_8_9 + b_2_2·b_2_32·a_1_1·a_3_5 − b_2_22·b_2_3·a_1_1·a_3_5
       + 2·b_2_23·a_1_1·a_3_5
  29. a_2_1·b_8_9 + b_2_2·b_2_32·a_1_1·a_3_5 − 2·b_2_22·b_2_3·a_1_1·a_3_5
       − b_2_23·a_1_0·a_3_5
  30. b_2_3·b_8_9 + b_2_2·b_2_34 − 2·b_2_22·b_2_33 − b_2_24·b_2_3
       + b_2_2·b_2_32·a_1_1·a_3_5 + b_2_22·b_2_3·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_5
  31. a_1_1·a_9_11 − b_2_33·a_1_1·a_3_5 − 2·b_2_2·b_2_32·a_1_1·a_3_5
       − 2·b_2_23·a_1_0·a_3_5
  32. a_1_0·a_9_11 − 2·b_2_22·b_2_3·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_4
  33. b_8_9·a_3_5 + b_2_2·b_2_33·a_3_5 − 2·b_2_22·b_2_32·a_3_5 − b_2_24·a_3_5
  34. b_8_9·a_3_4 − b_2_2·b_2_33·a_3_5 + b_2_22·b_2_32·a_3_5 − 2·b_2_23·b_2_3·a_3_5
       − 2·b_2_23·b_2_32·a_1_1 + b_2_24·b_2_3·a_1_1 − 2·b_2_25·a_1_1
  35. b_2_2·a_9_11 − 2·b_2_22·b_2_32·a_3_5 + b_2_22·b_2_33·a_1_1
       − 2·b_2_23·b_2_32·a_1_1 + 2·b_2_24·a_3_4 − b_2_24·b_2_3·a_1_1 − b_2_25·a_1_1
  36. a_2_0·a_9_11
  37. a_2_1·a_9_11
  38. b_2_3·a_9_11 − b_2_34·a_3_5 − 2·b_2_2·b_2_33·a_3_5 − 2·b_2_22·b_2_33·a_1_1
       − b_2_23·b_2_32·a_1_1 − 2·b_2_24·a_3_5 − 2·b_2_24·b_2_3·a_1_1
  39. a_3_5·a_9_11 + 2·b_2_22·b_2_32·a_1_1·a_3_5 + b_2_23·b_2_3·a_1_1·a_3_5
       + 2·b_2_24·a_1_1·a_3_5
  40. a_3_4·a_9_11 − 2·b_2_23·b_2_3·a_1_1·a_3_5 + b_2_24·a_1_1·a_3_5 + b_2_24·a_1_0·a_3_5
  41. b_8_9·a_7_8 − 2·b_2_23·b_2_33·a_3_5 − b_2_24·b_2_32·a_3_5
       + 2·b_2_24·b_2_33·a_1_1 + b_2_25·b_2_3·a_3_5 + b_2_25·b_2_32·a_1_1 + b_2_26·a_3_5
       − 2·b_2_26·b_2_3·a_1_1 − 2·b_2_27·a_1_1
  42. b_8_92 + 2·b_2_25·b_2_33 − b_2_27·b_2_3 − 2·b_2_24·b_2_32·a_1_1·a_3_5
       − 2·b_2_25·b_2_3·a_1_1·a_3_5 − 2·b_2_26·a_1_1·a_3_5 + 2·b_2_26·a_1_0·a_3_5
  43. a_7_8·a_9_11 + b_2_24·b_2_32·a_1_1·a_3_5 − b_2_25·b_2_3·a_1_1·a_3_5
  44. b_8_9·a_9_11 + 2·b_2_24·b_2_33·a_3_5 + b_2_25·b_2_32·a_3_5 + b_2_26·b_2_3·a_3_5
       − b_2_26·b_2_32·a_1_1 + b_2_27·a_3_5 + b_2_27·b_2_3·a_1_1

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 125

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 17.
  • 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_10_12, a Duflot regular element of degree 10
    2.  − b_8_9 + b_2_34 − b_2_2·b_2_33 + 2·b_2_22·b_2_32 + b_2_24, an element of degree 8
  • The Raw Filter Degree Type of that HSOP is [-1, 6, 16].
  • 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 125

Restriction maps

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. a_3_5 → 0, an element of degree 3
  9. a_7_8 → 0, an element of degree 7
  10. b_8_9 → 0, an element of degree 8
  11. a_9_11 → 0, an element of degree 9
  12. c_10_12 →  − c_2_05, an element of degree 10

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

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_4 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → 0, an element of degree 3
  9. a_7_8 → 2·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. b_8_9 →  − 2·c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 → 2·c_2_24·a_1_0 − 2·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. c_10_12 →  − 2·c_2_24·a_1_0·a_1_1 + c_2_1·c_2_24 − c_2_15, an element of degree 10

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. a_3_5 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  9. a_7_8 → 0, an element of degree 7
  10. b_8_9 → 0, an element of degree 8
  11. a_9_11 →  − c_2_24·a_1_0 + c_2_1·c_2_23·a_1_1, an element of degree 9
  12. c_10_12 →  − 2·c_2_24·a_1_0·a_1_1 + c_2_1·c_2_24 − c_2_15, an element of degree 10

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

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 → 2·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 →  − 2·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  9. a_7_8 → 2·c_2_23·a_1_1 + 2·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. b_8_9 → c_2_23·a_1_0·a_1_1 + 2·c_2_24, an element of degree 8
  11. a_9_11 → 0, an element of degree 9
  12. c_10_12 → 2·c_2_24·a_1_0·a_1_1 + c_2_1·c_2_24 − c_2_15, an element of degree 10

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

  1. a_1_0 → 2·a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 →  − 2·a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → 2·c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 →  − c_2_2·a_1_1 − 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  9. a_7_8 →  − 2·c_2_23·a_1_1 + 2·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. b_8_9 → 2·c_2_23·a_1_0·a_1_1 + 2·c_2_24, an element of degree 8
  11. a_9_11 →  − 2·c_2_24·a_1_0 + 2·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. c_10_12 → 2·c_2_24·a_1_0·a_1_1 − 2·c_2_25 + c_2_1·c_2_24 − c_2_15, an element of degree 10

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

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → 2·a_1_1, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → 2·a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → 2·c_2_2, an element of degree 2
  7. a_3_4 →  − c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → c_2_2·a_1_1 − 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1, an element of degree 3
  9. a_7_8 →  − c_2_23·a_1_1 + c_2_23·a_1_0 − c_2_1·c_2_22·a_1_1, an element of degree 7
  10. b_8_9 → c_2_23·a_1_0·a_1_1 + c_2_24, an element of degree 8
  11. a_9_11 → c_2_24·a_1_1 + c_2_24·a_1_0 − c_2_1·c_2_23·a_1_1, an element of degree 9
  12. c_10_12 → c_2_25 + c_2_1·c_2_24 − c_2_15, an element of degree 10

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

  1. a_1_0 →  − a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 → a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → a_1_0·a_1_1, an element of degree 2
  5. b_2_2 →  − c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 →  − 2·c_2_2·a_1_1 + c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → 2·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  9. a_7_8 →  − 2·c_2_23·a_1_1 − c_2_23·a_1_0 + c_2_1·c_2_22·a_1_1, an element of degree 7
  10. b_8_9 →  − 2·c_2_23·a_1_0·a_1_1 − c_2_24, an element of degree 8
  11. a_9_11 →  − c_2_24·a_1_0 + c_2_1·c_2_23·a_1_1, an element of degree 9
  12. c_10_12 →  − 2·c_2_24·a_1_0·a_1_1 − c_2_25 + c_2_1·c_2_24 − c_2_15, an element of degree 10

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 125


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