Mod-3-Cohomology of group number 86 of order 216

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General information on the group

  • The group order factors as 23 · 33.
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

The computation was based on 7 stability conditions for H*(E27; GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  −  2·t  +  2·t2  −  t3  +  2·t4  −  3·t5  +  2·t6  −  t7  +  2·t8  −  2·t9  +  t10
    ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (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].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 12:

  1. a_3_0, a nilpotent element of degree 3
  2. a_4_1, a nilpotent element of degree 4
  3. b_4_0, an element of degree 4
  4. a_5_0, a nilpotent element of degree 5
  5. a_9_1, a nilpotent element of degree 9
  6. b_10_0, an element of degree 10
  7. a_11_1, a nilpotent element of degree 11
  8. c_12_1, a Duflot element of degree 12

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Ring relations

There are 4 "obvious" relations:
   a_3_02, a_5_02, a_9_12, a_11_12

Apart from that, there are 16 minimal relations of maximal degree 21:

  1. a_4_1·a_3_0
  2. a_4_12
  3. a_4_1·b_4_0 − a_3_0·a_5_0
  4. a_4_1·a_5_0
  5. a_3_0·a_9_1
  6. a_4_1·a_9_1
  7. b_10_0·a_3_0 − b_4_0·a_9_1
  8. a_5_0·a_9_1 − a_3_0·a_11_1
  9. a_4_1·b_10_0 + a_3_0·a_11_1
  10. a_4_1·a_11_1
  11. b_10_0·a_5_0 + b_4_0·a_11_1
  12. a_5_0·a_11_1
  13. b_10_0·a_9_1 + b_4_0·c_12_1·a_3_0
  14. a_9_1·a_11_1 − c_12_1·a_3_0·a_5_0
  15. b_10_02 + b_4_02·c_12_1
  16. b_10_0·a_11_1 − b_4_0·c_12_1·a_5_0

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 21 using the Hilbert-Poincaré criterion.
  • 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_12_1, an element of degree 12
    2. b_4_0, an element of degree 4
  • A Duflot regular sequence is given by c_12_1.
  • 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*(E27; GF(3))

  1. a_3_0 → b_2_3·a_1_1 − b_2_1·a_1_1 + b_2_0·a_1_0
  2. a_4_1 → a_1_1·a_3_5 − a_1_0·a_3_4
  3. b_4_0 → b_2_32 − b_2_0·b_2_2 − b_2_0·b_2_1 + b_2_02 + a_1_0·a_3_5 − a_1_0·a_3_4
  4. a_5_0 → b_2_3·a_3_5 − b_2_0·a_3_4
  5. a_9_1 → b_2_03·b_2_1·a_1_1 − b_2_04·a_1_1 + b_2_3·c_6_8·a_1_1 + b_2_0·c_6_8·a_1_1
       − b_2_0·c_6_8·a_1_0
  6. b_10_0 → b_2_04·b_2_2 − b_2_03·a_1_1·a_3_5 − b_2_03·a_1_0·a_3_5 − b_2_03·a_1_0·a_3_4
       + b_2_32·c_6_8 + b_2_0·b_2_1·c_6_8 − b_2_02·c_6_8 + c_6_8·a_1_0·a_3_4
  7. a_11_1 → b_2_03·b_2_1·a_3_5 + b_2_04·a_3_5 − b_2_04·a_3_4 − b_2_04·b_2_1·a_1_1
       + b_2_05·a_1_1 − b_2_3·c_6_8·a_3_5 + b_2_1·c_6_8·a_3_5 − b_2_0·c_6_8·a_3_4
       − b_2_0·b_2_1·c_6_8·a_1_1 + b_2_02·c_6_8·a_1_1
  8. c_12_1 → b_2_05·b_2_2 − b_2_04·a_1_1·a_3_5 + b_2_04·a_1_0·a_3_5 + b_2_04·a_1_0·a_3_4
       − b_2_02·b_2_2·c_6_8 + b_2_0·c_6_8·a_1_1·a_3_5 + b_2_0·c_6_8·a_1_0·a_3_5
       + b_2_0·c_6_8·a_1_0·a_3_4 − c_6_82

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_3_0 → 0, an element of degree 3
  2. a_4_1 → 0, an element of degree 4
  3. b_4_0 → 0, an element of degree 4
  4. a_5_0 → 0, an element of degree 5
  5. a_9_1 → 0, an element of degree 9
  6. b_10_0 → 0, an element of degree 10
  7. a_11_1 → 0, an element of degree 11
  8. c_12_1 →  − c_2_06, an element of degree 12

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

  1. a_3_0 → c_2_2·a_1_1, an element of degree 3
  2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0 → c_2_22, an element of degree 4
  4. a_5_0 → c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1 →  − c_2_1·c_2_23·a_1_1 + c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0 →  − c_2_1·c_2_24 + c_2_13·c_2_22, an element of degree 10
  7. a_11_1 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  8. c_12_1 →  − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12

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

  1. a_3_0 → c_2_2·a_1_1, an element of degree 3
  2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0 → c_2_22, an element of degree 4
  4. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1 → c_2_1·c_2_23·a_1_1 − c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0 → c_2_1·c_2_24 − c_2_13·c_2_22, an element of degree 10
  7. a_11_1 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  8. c_12_1 →  − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12

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

  1. a_3_0 → c_2_2·a_1_1, an element of degree 3
  2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0 → c_2_22, an element of degree 4
  4. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1 → c_2_1·c_2_23·a_1_1 − c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0 → c_2_1·c_2_24 − c_2_13·c_2_22, an element of degree 10
  7. a_11_1 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  8. c_12_1 →  − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12

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

  1. a_3_0 → c_2_2·a_1_1, an element of degree 3
  2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0 → c_2_22, an element of degree 4
  4. a_5_0 → c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_1 →  − c_2_1·c_2_23·a_1_1 + c_2_13·c_2_2·a_1_1, an element of degree 9
  6. b_10_0 →  − c_2_1·c_2_24 + c_2_13·c_2_22, an element of degree 10
  7. a_11_1 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  8. c_12_1 →  − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, 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