Mod-3-Cohomology of group number 1027 of order 288

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General information on the group

  • The group order factors as 25 · 32.
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 2.
  • Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

The computation was based on 15 stability conditions for H*(SmallGroup(9,2); GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1  −  2·t  +  3·t2  −  4·t3  +  4·t4  −  4·t5  +  4·t6  −  3·t7  +  3·t8  −  3·t9  +  4·t10  −  4·t11  +  4·t12  −  4·t13  +  3·t14  −  2·t15  +  t16
    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4) · (1  +  t8)
  • 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 16:

  1. a_7_0, a nilpotent element of degree 7
  2. c_8_0, a Duflot element of degree 8
  3. a_10_0, a nilpotent element of degree 10
  4. a_11_1, a nilpotent element of degree 11
  5. a_11_0, a nilpotent element of degree 11
  6. c_12_0, a Duflot element of degree 12
  7. a_15_1, a nilpotent element of degree 15
  8. c_16_1, a Duflot element of degree 16

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Ring relations

There are 4 "obvious" relations:
   a_7_02, a_11_02, a_11_12, a_15_12

Apart from that, there are 16 minimal relations of maximal degree 27:

  1. a_10_0·a_7_0
  2. a_7_0·a_11_0 − c_8_0·a_10_0
  3. a_7_0·a_11_1 + c_8_0·a_10_0
  4. c_12_0·a_7_0 − c_8_0·a_11_1 − c_8_0·a_11_0
  5. a_10_02
  6. a_10_0·a_11_0
  7. a_10_0·a_11_1
  8. a_7_0·a_15_1 + a_10_0·c_12_0
  9. a_11_0·a_11_1 + a_10_0·c_12_0
  10. c_12_0·a_11_0 + c_8_0·a_15_1 − c_8_02·a_7_0
  11. c_16_1·a_7_0 + c_12_0·a_11_1 − c_8_0·a_15_1
  12. c_12_02 + c_8_0·c_16_1 − c_8_03
  13. a_10_0·a_15_1
  14. a_11_0·a_15_1 + c_8_02·a_10_0
  15. a_11_1·a_15_1 − a_10_0·c_16_1
  16. c_16_1·a_11_0 − c_12_0·a_15_1 + c_8_02·a_11_1

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 27 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_8_0, an element of degree 8
    2. c_16_1, an element of degree 16
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 22].

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(9,2); GF(3))

  1. a_7_0 → c_2_23·a_1_1 + c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1 + c_2_13·a_1_0
  2. c_8_0 → c_2_24 − c_2_12·c_2_22 + c_2_14
  3. a_10_0 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1
  4. 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
  5. a_11_0 → c_2_25·a_1_1 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0 + c_2_15·a_1_0
  6. c_12_0 → c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16
  7. a_15_1 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1
  8. c_16_1 → c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_7_0 → c_2_23·a_1_1 + c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1 + c_2_13·a_1_0, an element of degree 7
  2. c_8_0 → c_2_24 − c_2_12·c_2_22 + c_2_14, an element of degree 8
  3. a_10_0 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  4. 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
  5. a_11_0 → c_2_25·a_1_1 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0 + c_2_15·a_1_0, an element of degree 11
  6. c_12_0 → c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
  7. a_15_1 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  8. c_16_1 → c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16

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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