Cohomology of group number 3 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 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) · (t2  +  1)
    (t  +  1) · (t  −  1)3 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)

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Ring generators

The cohomology ring has 16 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. c_2_3, a Duflot regular 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_4_4, a nilpotent element of degree 4
  10. a_5_9, a nilpotent element of degree 5
  11. a_6_7, a nilpotent element of degree 6
  12. a_7_13, a nilpotent element of degree 7
  13. a_8_10, a nilpotent element of degree 8
  14. a_9_17, a nilpotent element of degree 9
  15. a_10_13, a nilpotent element of degree 10
  16. c_10_19, 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_3_42, a_3_52, a_5_92, a_7_132, a_9_172

Apart from that, there are 83 minimal relations of maximal degree 20:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_0
  3. a_2_1·a_1_1 − 2·a_2_0·a_1_1
  4. a_2_1·a_1_0 − a_2_0·a_1_1
  5. b_2_2·a_1_0
  6. a_2_02
  7. a_2_12
  8. a_2_0·a_2_1
  9.  − a_2_0·b_2_2 + a_1_1·a_3_4
  10. a_1_0·a_3_4
  11.  − a_2_1·b_2_2 + a_2_0·b_2_2 + a_1_1·a_3_5
  12. 2·a_2_0·b_2_2 + a_1_0·a_3_5
  13. a_2_0·a_3_4
  14. b_2_2·a_3_4
  15. a_2_1·a_3_5 + 2·a_2_1·a_3_4
  16.  − a_2_1·a_3_4 + a_2_0·a_3_5
  17. a_4_4·a_1_1 − a_2_1·a_3_4 + 2·a_2_0·c_2_3·a_1_1
  18. a_4_4·a_1_0
  19. a_2_1·a_4_4
  20. a_2_0·a_4_4
  21. b_2_2·a_4_4 + 2·a_3_4·a_3_5 − b_2_2·a_1_1·a_3_5 − c_2_3·a_1_0·a_3_5
  22. 2·a_3_4·a_3_5 + a_1_1·a_5_9 − c_2_3·a_1_0·a_3_5
  23. a_1_0·a_5_9
  24. a_4_4·a_3_4
  25. 2·a_4_4·a_3_5 + a_2_1·a_5_9 + a_2_0·c_2_3·a_3_5
  26. a_2_0·a_5_9
  27. b_2_2·a_5_9 − 2·b_2_22·c_2_3·a_1_1
  28. a_6_7·a_1_1 + 2·a_4_4·a_3_5 + a_2_0·c_2_3·a_3_5 − a_2_0·c_2_32·a_1_1
  29. a_6_7·a_1_0
  30. a_4_42
  31. a_3_4·a_5_9
  32. a_2_1·a_6_7
  33. a_2_0·a_6_7
  34. b_2_2·a_6_7 + a_3_5·a_5_9 + 2·b_2_22·a_1_1·a_3_5 − 2·c_2_3·a_1_1·a_5_9
  35. a_3_5·a_5_9 + a_1_1·a_7_13 + b_2_2·c_2_3·a_1_1·a_3_5 − 2·c_2_32·a_1_1·a_3_5
       − 2·c_2_32·a_1_0·a_3_5
  36. a_1_0·a_7_13 − 2·c_2_32·a_1_0·a_3_5
  37. a_4_4·a_5_9
  38. a_6_7·a_3_4
  39.  − 2·a_6_7·a_3_5 + a_2_1·a_7_13 + a_2_1·c_2_3·a_5_9 + 2·a_2_0·c_2_32·a_3_5
       + 2·a_2_0·c_2_33·a_1_1
  40. a_2_0·a_7_13 − 2·a_2_0·c_2_32·a_3_5 + a_2_0·c_2_33·a_1_1
  41. b_2_2·a_7_13 − b_2_22·c_2_3·a_3_5 + 2·b_2_23·c_2_3·a_1_1 − 2·b_2_2·c_2_32·a_3_5
       − 2·b_2_22·c_2_32·a_1_1 + b_2_2·c_2_33·a_1_1
  42. a_8_10·a_1_1 − 2·a_6_7·a_3_5 + 2·a_2_1·c_2_3·a_5_9 − 2·a_2_0·c_2_32·a_3_5
       + 2·a_2_0·c_2_33·a_1_1
  43. a_8_10·a_1_0 + 2·a_2_0·c_2_33·a_1_1
  44. a_4_4·a_6_7
  45. a_3_4·a_7_13 + c_2_32·a_1_1·a_5_9 + 2·c_2_33·a_1_0·a_3_5
  46. a_3_5·a_7_13 + c_2_3·a_1_1·a_7_13 − 2·b_2_22·c_2_3·a_1_1·a_3_5 + c_2_32·a_1_1·a_5_9
       + b_2_2·c_2_32·a_1_1·a_3_5 + 2·c_2_33·a_1_1·a_3_5 + 2·c_2_33·a_1_0·a_3_5
  47. a_2_1·a_8_10
  48. a_2_0·a_8_10
  49.  − b_2_2·a_8_10 − a_3_5·a_7_13 + a_1_1·a_9_17 − c_2_32·a_1_1·a_5_9
       + 2·c_2_33·a_1_1·a_3_5
  50. a_1_0·a_9_17 − 2·c_2_33·a_1_0·a_3_5
  51. a_6_7·a_5_9
  52. a_4_4·a_7_13 + a_2_1·c_2_32·a_5_9 + 2·a_2_0·c_2_33·a_3_5 − 2·a_2_0·c_2_34·a_1_1
  53. a_8_10·a_3_4 + 2·a_2_0·c_2_33·a_3_5
  54. a_8_10·a_3_5 + a_2_1·a_9_17 + a_2_1·c_2_3·a_7_13 + a_2_1·c_2_32·a_5_9
       + 2·a_2_0·c_2_34·a_1_1
  55. a_2_0·a_9_17 − 2·a_2_0·c_2_33·a_3_5
  56. a_10_13·a_1_1 + a_8_10·a_3_5 + a_2_1·c_2_3·a_7_13 − a_2_0·c_2_33·a_3_5
       − 2·a_2_0·c_2_34·a_1_1
  57. a_10_13·a_1_0
  58. a_6_72
  59. 2·a_5_9·a_7_13 + c_2_32·a_1_1·a_7_13 + b_2_22·c_2_32·a_1_1·a_3_5
       − 2·c_2_33·a_1_1·a_5_9 + b_2_2·c_2_33·a_1_1·a_3_5 − 2·c_2_34·a_1_1·a_3_5
       − 2·c_2_34·a_1_0·a_3_5
  60. a_4_4·a_8_10
  61. a_3_4·a_9_17 + c_2_33·a_1_1·a_5_9 − c_2_34·a_1_0·a_3_5
  62. a_2_1·a_10_13
  63. a_2_0·a_10_13
  64. b_2_2·a_10_13 − 2·a_5_9·a_7_13 − a_3_5·a_9_17 + 2·b_2_24·a_1_1·a_3_5
       + c_2_3·a_1_1·a_9_17 + 2·b_2_23·c_2_3·a_1_1·a_3_5 + b_2_2·c_2_33·a_1_1·a_3_5
       − 2·c_2_34·a_1_1·a_3_5 − 2·c_2_34·a_1_0·a_3_5
  65. a_6_7·a_7_13 − a_2_1·c_2_32·a_7_13 − 2·a_2_0·c_2_34·a_3_5 − a_2_0·c_2_35·a_1_1
  66. a_8_10·a_5_9 + 2·a_2_1·c_2_33·a_5_9
  67.  − a_4_4·a_9_17 + a_1_1·a_3_5·a_9_17 − a_2_1·c_2_33·a_5_9 − a_2_0·c_2_34·a_3_5
  68. a_10_13·a_3_5 − a_2_1·c_2_3·a_9_17 − a_2_0·c_10_19·a_1_1 + a_2_1·c_2_33·a_5_9
       + 2·a_2_0·c_2_34·a_3_5
  69. a_10_13·a_3_4
  70. a_6_7·a_8_10
  71. 2·a_5_9·a_9_17 + b_2_2·c_2_3·a_1_1·a_9_17 + c_2_33·a_1_1·a_7_13
       − b_2_2·c_2_34·a_1_1·a_3_5 − 2·c_2_35·a_1_1·a_3_5 − 2·c_2_35·a_1_0·a_3_5
  72. a_4_4·a_10_13
  73.  − 2·a_8_10·a_7_13 − 2·c_2_3·a_1_1·a_3_5·a_9_17 + a_2_1·c_2_32·a_9_17
       − 2·a_2_1·c_2_34·a_5_9 + a_2_0·c_2_35·a_3_5 − a_2_0·c_2_36·a_1_1
  74. a_6_7·a_9_17 + 2·b_2_2·a_1_1·a_3_5·a_9_17 − 2·c_2_3·a_1_1·a_3_5·a_9_17
       − a_2_1·c_2_33·a_7_13 − a_2_1·c_2_34·a_5_9 − 2·a_2_0·c_2_35·a_3_5
       − 2·a_2_0·c_2_36·a_1_1
  75. a_10_13·a_5_9 − 2·c_2_3·a_1_1·a_3_5·a_9_17
  76. a_8_102
  77. a_7_13·a_9_17 − b_2_2·c_2_3·a_3_5·a_9_17 + 2·b_2_22·c_2_3·a_1_1·a_9_17
       − 2·c_2_32·a_3_5·a_9_17 − 2·b_2_2·c_2_32·a_1_1·a_9_17 + c_2_33·a_1_1·a_9_17
       − 2·c_2_34·a_1_1·a_7_13 − 2·c_2_35·a_1_1·a_5_9 + 2·b_2_2·c_2_35·a_1_1·a_3_5
       − c_2_36·a_1_1·a_3_5 + c_2_36·a_1_0·a_3_5
  78. a_6_7·a_10_13
  79. a_8_10·a_9_17 + 2·b_2_2·c_2_3·a_1_1·a_3_5·a_9_17 − 2·c_2_32·a_1_1·a_3_5·a_9_17
       − a_2_1·c_2_33·a_9_17 + 2·a_2_1·c_2_34·a_7_13 + 2·a_2_1·c_2_35·a_5_9
       − 2·a_2_0·c_2_36·a_3_5 − a_2_0·c_2_37·a_1_1
  80. a_10_13·a_7_13 + 2·b_2_2·c_2_3·a_1_1·a_3_5·a_9_17 + 2·c_2_32·a_1_1·a_3_5·a_9_17
       − a_2_1·c_2_33·a_9_17 − 2·a_2_0·c_2_32·c_10_19·a_1_1 − 2·a_2_1·c_2_35·a_5_9
       − a_2_0·c_2_37·a_1_1
  81. a_8_10·a_10_13
  82. a_10_13·a_9_17 + 2·b_2_23·a_1_1·a_3_5·a_9_17 + 2·b_2_22·c_2_3·a_1_1·a_3_5·a_9_17
       + b_2_2·c_2_32·a_1_1·a_3_5·a_9_17 − 2·c_2_33·a_1_1·a_3_5·a_9_17
       − 2·a_2_1·c_2_34·a_9_17 − 2·a_2_0·c_2_33·c_10_19·a_1_1 + 2·a_2_1·c_2_36·a_5_9
       − a_2_0·c_2_37·a_3_5
  83. a_10_132

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 20.
  • 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_3, a Duflot regular element of degree 2
    2. c_10_19, a Duflot regular element of degree 10
    3. b_2_2, 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_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. c_2_3 → c_2_2, 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_4_4 → 0, an element of degree 4
  10. a_5_9 → 0, an element of degree 5
  11. a_6_7 → 0, an element of degree 6
  12. a_7_13 → 0, an element of degree 7
  13. a_8_10 → 0, an element of degree 8
  14. a_9_17 → 0, an element of degree 9
  15. a_10_13 → 0, an element of degree 10
  16. c_10_19 →  − c_2_15, 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_2_0 → 0, an element of degree 2
  4. a_2_1 → a_1_0·a_1_2, an element of degree 2
  5. b_2_2 → c_2_5, an element of degree 2
  6. c_2_3 → c_2_4, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. a_3_5 →  − c_2_5·a_1_0 + c_2_3·a_1_2, an element of degree 3
  9. a_4_4 → c_2_5·a_1_0·a_1_2, an element of degree 4
  10. a_5_9 → 2·c_2_4·c_2_5·a_1_2, an element of degree 5
  11. a_6_7 →  − 2·c_2_52·a_1_0·a_1_2 + 2·c_2_4·c_2_5·a_1_0·a_1_2, an element of degree 6
  12. a_7_13 →  − 2·c_2_4·c_2_52·a_1_2 − c_2_4·c_2_52·a_1_0 + 2·c_2_42·c_2_5·a_1_2
       − 2·c_2_42·c_2_5·a_1_0 − c_2_43·a_1_2 + c_2_3·c_2_4·c_2_5·a_1_2
       + 2·c_2_3·c_2_42·a_1_2, an element of degree 7
  13. a_8_10 →  − c_2_53·a_1_1·a_1_2 + c_2_53·a_1_0·a_1_2 + 2·c_2_4·c_2_52·a_1_0·a_1_2
       + c_2_42·c_2_5·a_1_0·a_1_2 − 2·c_2_43·a_1_0·a_1_2, an element of degree 8
  14. a_9_17 → c_2_54·a_1_1 − c_2_54·a_1_0 + c_2_4·c_2_53·a_1_2 + c_2_4·c_2_53·a_1_0
       + 2·c_2_42·c_2_52·a_1_2 + c_2_42·c_2_52·a_1_0 − c_2_43·c_2_5·a_1_2
       − 2·c_2_43·c_2_5·a_1_0 + c_2_3·c_2_53·a_1_2 − c_2_3·c_2_4·c_2_52·a_1_2
       − c_2_3·c_2_42·c_2_5·a_1_2 + 2·c_2_3·c_2_43·a_1_2, an element of degree 9
  15. a_10_13 →  − 2·c_2_54·a_1_0·a_1_2 − c_2_54·a_1_0·a_1_1 + c_2_4·c_2_53·a_1_1·a_1_2
       + c_2_4·c_2_53·a_1_0·a_1_2 − 2·c_2_42·c_2_52·a_1_0·a_1_2
       − c_2_43·c_2_5·a_1_0·a_1_2 − c_2_3·c_2_53·a_1_1·a_1_2, an element of degree 10
  16. c_10_19 → c_2_54·a_1_0·a_1_2 − 2·c_2_54·a_1_0·a_1_1 − 2·c_2_4·c_2_53·a_1_0·a_1_2
       − c_2_42·c_2_52·a_1_0·a_1_2 + c_2_43·c_2_5·a_1_0·a_1_2 + c_2_44·a_1_0·a_1_2
       − 2·c_2_3·c_2_53·a_1_1·a_1_2 − 2·c_2_4·c_2_54 + c_2_42·c_2_53
       − 2·c_2_43·c_2_52 − c_2_44·c_2_5 + c_2_3·c_2_54 − c_2_35, 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