Cohomology of group number 55 of order 243

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General information on the group

  • The group has 3 minimal generators and exponent 9.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 4 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2, 2, 2 and 3, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    t3  −  t2  −  1
    (t  −  1)3 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 7:

  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_2, an element of degree 2
  5. b_2_3, an element of degree 2
  6. b_2_4, an element of degree 2
  7. b_2_5, an element of degree 2
  8. a_3_8, a nilpotent element of degree 3
  9. b_4_10, an element of degree 4
  10. a_5_12, a nilpotent element of degree 5
  11. b_6_14, an element of degree 6
  12. c_6_15, a Duflot regular element of degree 6
  13. a_7_19, a nilpotent element of degree 7

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Ring relations

There are 6 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_82, a_5_122, a_7_192

Apart from that, there are 39 minimal relations of maximal degree 13:

  1. a_1_0·a_1_1
  2. b_2_3·a_1_0 − b_2_2·a_1_1
  3. b_2_4·a_1_1 − b_2_2·a_1_1
  4. b_2_4·a_1_0 − b_2_3·a_1_1
  5. b_2_5·a_1_0 − b_2_2·a_1_1
  6. b_2_3·b_2_4 − b_2_2·b_2_3
  7.  − b_2_32 + b_2_2·b_2_4
  8. b_2_42 − b_2_32
  9. b_2_3·b_2_5 − b_2_32
  10. b_2_2·b_2_5 − b_2_2·b_2_3
  11.  − b_2_4·b_2_5 + b_2_2·b_2_3 + a_1_1·a_3_8
  12. a_1_0·a_3_8
  13. b_2_3·a_3_8 − b_2_2·b_2_3·a_1_1 + b_2_22·a_1_1
  14. b_2_2·a_3_8 + b_2_2·b_2_3·a_1_1 − b_2_22·a_1_1
  15. b_2_4·a_3_8 + b_2_2·b_2_3·a_1_1 − b_2_22·a_1_1
  16. b_4_10·a_1_0 − b_2_2·b_2_3·a_1_1
  17. b_2_3·b_4_10 − b_2_22·b_2_3
  18. b_2_2·b_4_10 − b_2_22·b_2_4
  19.  − b_2_4·b_4_10 + b_2_22·b_2_4 + a_1_1·a_5_12 − b_2_5·a_1_1·a_3_8
  20. a_1_0·a_5_12
  21. b_2_3·a_5_12 + b_2_22·b_2_3·a_1_1 − b_2_23·a_1_1
  22. b_2_2·a_5_12 − b_2_22·b_2_3·a_1_1 + b_2_23·a_1_1
  23. b_2_4·a_5_12 − b_2_22·b_2_3·a_1_1 + b_2_23·a_1_1
  24. b_6_14·a_1_1 + b_4_10·a_3_8 − b_2_5·a_5_12 + b_2_5·b_4_10·a_1_1 + b_2_52·a_3_8
       + b_2_22·b_2_3·a_1_1
  25. b_6_14·a_1_0 + b_2_22·b_2_3·a_1_1 + b_2_23·a_1_1
  26. b_2_3·b_6_14 + b_2_23·b_2_4 + b_2_23·b_2_3
  27. b_2_2·b_6_14 + b_2_23·b_2_4 + b_2_23·b_2_3
  28. b_2_4·b_6_14 + b_2_23·b_2_4 + b_2_23·b_2_3 + a_3_8·a_5_12 + b_2_5·a_1_1·a_5_12
       − b_2_52·a_1_1·a_3_8
  29. a_3_8·a_5_12 + a_1_1·a_7_19 + b_2_5·a_1_1·a_5_12 + b_2_52·a_1_1·a_3_8
  30. a_1_0·a_7_19
  31. b_2_3·a_7_19 − b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1 − b_2_2·c_6_15·a_1_1
  32. b_2_2·a_7_19 − b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1 − b_2_2·c_6_15·a_1_0
  33.  − b_6_14·a_3_8 + b_4_102·a_1_1 + b_2_5·a_7_19 + b_2_5·b_4_10·a_3_8 − b_2_52·a_5_12
       − b_2_52·b_4_10·a_1_1 − b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1 − b_2_2·c_6_15·a_1_1
  34. b_2_4·a_7_19 − b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1 − b_2_3·c_6_15·a_1_1
  35. a_3_8·a_7_19 − b_4_10·a_1_1·a_5_12 + b_2_5·a_1_1·a_7_19 − b_2_53·a_1_1·a_3_8
  36.  − b_6_14·a_5_12 + b_4_10·a_7_19 − b_4_102·a_3_8 + b_2_5·b_4_10·a_5_12
       + b_2_5·b_4_102·a_1_1 + b_2_52·a_7_19 + b_2_53·a_5_12 + b_2_53·b_4_10·a_1_1
       + b_2_54·a_3_8 + b_2_24·b_2_3·a_1_1 − b_2_25·a_1_1 − b_2_52·c_6_15·a_1_1
       + b_2_2·b_2_3·c_6_15·a_1_1 + b_2_22·c_6_15·a_1_1
  37. b_6_142 + b_4_103 − b_2_5·b_4_10·b_6_14 + b_2_53·b_6_14 − b_2_54·b_4_10
       + b_2_25·b_2_4 + b_2_25·b_2_3 + b_2_52·a_1_1·a_7_19 − b_2_53·a_1_1·a_5_12
       + b_2_53·c_6_15 − b_2_22·b_2_3·c_6_15
  38.  − b_6_142 − b_4_103 + b_2_5·b_4_10·b_6_14 − b_2_53·b_6_14 + b_2_54·b_4_10
       − b_2_25·b_2_4 − b_2_25·b_2_3 + a_5_12·a_7_19 − b_4_10·a_1_1·a_7_19
       + b_2_5·b_4_10·a_1_1·a_5_12 + b_2_53·a_1_1·a_5_12 − b_2_54·a_1_1·a_3_8
       − b_2_53·c_6_15 + b_2_22·b_2_3·c_6_15 + b_2_5·c_6_15·a_1_1·a_3_8
  39.  − b_6_14·a_7_19 − b_4_102·a_5_12 + b_4_103·a_1_1 + b_2_5·b_4_10·a_7_19
       − b_2_5·b_4_102·a_3_8 − b_2_52·b_4_10·a_5_12 − b_2_52·b_4_102·a_1_1
       − b_2_53·b_4_10·a_3_8 − b_2_54·a_5_12 − b_2_54·b_4_10·a_1_1 + b_2_55·a_3_8
       + b_2_25·b_2_3·a_1_1 − b_2_52·c_6_15·a_3_8 − b_2_53·c_6_15·a_1_1
       − b_2_22·b_2_3·c_6_15·a_1_1 − b_2_23·c_6_15·a_1_1

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 13.
  • 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_6_15, a Duflot regular element of degree 6
    2. b_4_103 − b_2_52·b_4_102 + b_2_53·b_6_14 − b_2_54·b_4_10 − b_2_56 + b_2_25·b_2_3
         − b_2_26 + b_2_53·c_6_15 − b_2_22·b_2_3·c_6_15, an element of degree 12
    3. b_2_52·b_4_103 − b_2_54·b_4_102 + b_2_55·b_6_14 − b_2_56·b_4_10 − b_2_27·b_2_4
         + b_2_27·b_2_3 + b_2_55·c_6_15 − b_2_24·b_2_3·c_6_15, an element of degree 16
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 15, 31].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 4.

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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_1_2 → 0, an element of degree 1
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_2_4 → 0, an element of degree 2
  7. b_2_5 → 0, an element of degree 2
  8. a_3_8 → 0, an element of degree 3
  9. b_4_10 → 0, an element of degree 4
  10. a_5_12 → 0, an element of degree 5
  11. b_6_14 → 0, an element of degree 6
  12. c_6_15 → c_2_03, an element of degree 6
  13. a_7_19 → 0, an element of degree 7

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_1_2 → 0, an element of degree 1
  4. b_2_2 → c_2_2, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_2_4 → 0, an element of degree 2
  7. b_2_5 → 0, an element of degree 2
  8. a_3_8 → 0, an element of degree 3
  9. b_4_10 → 0, an element of degree 4
  10. a_5_12 → 0, an element of degree 5
  11. b_6_14 → 0, an element of degree 6
  12. c_6_15 →  − c_2_1·c_2_22 + c_2_13, an element of degree 6
  13. a_7_19 →  − c_2_1·c_2_22·a_1_1 + c_2_13·a_1_1, an element of degree 7

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_1_2 → a_1_1, an element of degree 1
  4. b_2_2 → c_2_2, an element of degree 2
  5. b_2_3 →  − c_2_2, an element of degree 2
  6. b_2_4 → c_2_2, an element of degree 2
  7. b_2_5 →  − c_2_2, an element of degree 2
  8. a_3_8 → c_2_2·a_1_1, an element of degree 3
  9. b_4_10 → c_2_22, an element of degree 4
  10. a_5_12 →  − c_2_22·a_1_1, an element of degree 5
  11. b_6_14 → 0, an element of degree 6
  12. c_6_15 →  − c_2_23 − c_2_1·c_2_22 + c_2_13, an element of degree 6
  13. a_7_19 →  − c_2_23·a_1_1 − c_2_1·c_2_22·a_1_1 + c_2_13·a_1_1, an element of degree 7

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_1_2 →  − a_1_1, an element of degree 1
  4. b_2_2 → c_2_2, an element of degree 2
  5. b_2_3 → c_2_2, an element of degree 2
  6. b_2_4 → c_2_2, an element of degree 2
  7. b_2_5 → c_2_2, an element of degree 2
  8. a_3_8 → 0, an element of degree 3
  9. b_4_10 → c_2_22, an element of degree 4
  10. a_5_12 → 0, an element of degree 5
  11. b_6_14 → c_2_23, an element of degree 6
  12. c_6_15 →  − c_2_1·c_2_22 + c_2_13, an element of degree 6
  13. a_7_19 →  − c_2_23·a_1_1 − c_2_1·c_2_22·a_1_1 + c_2_13·a_1_1, an element of degree 7

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_1, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_2_4 →  − a_1_1·a_1_2, an element of degree 2
  7. b_2_5 → c_2_4, an element of degree 2
  8. a_3_8 → c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
  9. b_4_10 →  − c_2_4·a_1_1·a_1_2 − c_2_52 + c_2_4·c_2_5 − c_2_3·c_2_4, an element of degree 4
  10. a_5_12 → c_2_52·a_1_2 + c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2 − c_2_42·a_1_2 − c_2_3·c_2_5·a_1_1
       + c_2_3·c_2_4·a_1_2, an element of degree 5
  11. b_6_14 → c_2_53 + c_2_4·c_2_52 + c_2_42·c_2_5 + c_2_3·c_2_42, an element of degree 6
  12. c_6_15 → c_2_52·a_1_1·a_1_2 − c_2_4·c_2_5·a_1_1·a_1_2 − c_2_42·a_1_1·a_1_2
       + c_2_3·c_2_4·a_1_1·a_1_2 − c_2_53 + c_2_4·c_2_52 − c_2_3·c_2_52
       + c_2_3·c_2_4·c_2_5 + c_2_3·c_2_42 + c_2_32·c_2_4 + c_2_33, an element of degree 6
  13. a_7_19 →  − c_2_53·a_1_2 + c_2_53·a_1_1 − c_2_4·c_2_52·a_1_2 − c_2_4·c_2_52·a_1_1
       − c_2_42·c_2_5·a_1_2 + c_2_42·c_2_5·a_1_1 − c_2_43·a_1_2 + c_2_3·c_2_52·a_1_1
       − c_2_3·c_2_42·a_1_2 − c_2_3·c_2_42·a_1_1 − c_2_32·c_2_4·a_1_1, an element of degree 7

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243


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