Mod-2-Cohomology of SymmetricGroup(9), a group of order 362880

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General information on the group

  • SymmetricGroup(9) is a group of order 362880.
  • The group order factors as 27 · 34 · 5 · 7.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9),(1,2)]).
  • It is non-abelian.
  • It has 2-Rank 4.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 4, 4 and 4, respectively.

Structure of the cohomology ring

The computation was based on 5 stability conditions for H*(SmallGroup(384,5602); GF(2)).

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  +  2·t2  +  t3  +  3·t4  +  3·t5  +  4·t6  +  4·t7  +  4·t8  +  3·t9  +  3·t10  +  3·t11  +  t12  +  2·t13  +  t15
    (1  +  t) · ( − 1  +  t)4 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2)2 · (1  +  t  +  t2  +  t3  +  t4  +  t5  +  t6)
  • The a-invariants are -∞,-∞,-∞,-10,-4. 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, -3, -4, -4].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 7:

  1. b_1_0, an element of degree 1
  2. b_2_1, an element of degree 2
  3. b_3_1, an element of degree 3
  4. b_3_0, an element of degree 3
  5. c_4_3, a Duflot element of degree 4
  6. b_5_3, an element of degree 5
  7. b_6_0, an element of degree 6
  8. b_7_18, an element of degree 7

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Ring relations

There are 10 minimal relations of maximal degree 12:

  1. b_3_0·b_3_1 + b_1_0·b_5_3 + b_1_03·b_3_1 + b_2_1·b_1_0·b_3_1
  2. b_6_0·b_1_0
  3. b_1_0·b_7_18
  4. b_3_0·b_5_3 + b_1_03·b_5_3 + b_1_05·b_3_1 + b_2_1·b_1_03·b_3_1 + c_4_3·b_1_0·b_3_1
  5. b_2_1·b_7_18
  6. b_6_0·b_3_0
  7. b_3_0·b_7_18
  8. b_3_1·b_7_18
  9. b_5_32 + b_1_04·b_3_12 + b_2_1·b_3_1·b_5_3 + b_2_1·b_1_02·b_3_12 + b_2_12·b_6_0
       + c_4_3·b_3_12
  10. b_5_3·b_7_18

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 26 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 12 and the last generator in degree 7.
  • The following is a filter regular homogeneous system of parameters:
    1. b_3_1·b_5_3 + b_1_02·b_3_02 + b_1_08 + b_2_1·b_1_0·b_5_3 + b_2_1·b_1_03·b_3_1
         + b_2_14 + c_4_3·b_1_0·b_3_0 + b_2_1·c_4_3·b_1_02 + c_4_32, an element of degree 8
    2. b_3_14 + b_3_04 + b_1_0·b_3_12·b_5_3 + b_1_03·b_3_13 + b_1_04·b_3_1·b_5_3
         + b_1_06·b_3_02 + b_6_0·b_3_12 + b_6_02 + b_2_1·b_1_05·b_5_3
         + b_2_1·b_1_07·b_3_1 + b_2_12·b_1_02·b_3_12 + b_2_12·b_1_02·b_3_02
         + b_2_13·b_1_0·b_5_3 + b_2_13·b_1_03·b_3_1 + b_2_13·b_6_0 + b_2_14·b_1_04
         + c_4_3·b_3_1·b_5_3 + c_4_3·b_1_02·b_3_12 + c_4_3·b_1_02·b_3_02
         + c_4_3·b_1_03·b_5_3 + c_4_3·b_1_05·b_3_1 + c_4_3·b_1_05·b_3_0
         + b_2_1·c_4_3·b_3_12 + b_2_1·c_4_3·b_1_0·b_5_3 + b_2_1·c_4_3·b_1_03·b_3_0
         + b_2_1·c_4_3·b_1_06 + b_2_12·c_4_3·b_1_0·b_3_0 + b_2_13·c_4_3·b_1_02
         + c_4_32·b_1_0·b_3_0 + b_2_12·c_4_32, an element of degree 12
    3. b_7_182 + b_3_13·b_5_3 + b_1_02·b_3_04 + b_1_03·b_3_12·b_5_3 + b_1_05·b_3_13
         + b_6_0·b_3_1·b_5_3 + b_2_1·b_1_0·b_3_12·b_5_3 + b_2_1·b_1_03·b_3_13
         + b_2_1·b_6_0·b_3_12 + b_2_12·b_1_04·b_3_12 + b_2_12·b_1_04·b_3_02
         + b_2_13·b_3_1·b_5_3 + b_2_13·b_1_02·b_3_12 + b_2_13·b_1_03·b_5_3
         + b_2_13·b_1_05·b_3_1 + b_2_14·b_6_0 + c_4_3·b_1_0·b_3_13 + c_4_3·b_1_0·b_3_03
         + c_4_3·b_1_04·b_3_02 + c_4_3·b_1_05·b_5_3 + c_4_3·b_1_07·b_3_1
         + b_2_1·c_4_3·b_1_03·b_5_3 + b_2_1·c_4_3·b_1_05·b_3_0 + b_2_12·c_4_3·b_3_12
         + b_2_12·c_4_3·b_1_0·b_5_3 + b_2_12·c_4_3·b_1_03·b_3_1 + b_2_12·c_4_3·b_6_0
         + b_2_13·c_4_3·b_1_04 + c_4_32·b_3_02 + c_4_32·b_1_03·b_3_1 + c_4_32·b_1_06
         + b_2_1·c_4_32·b_1_0·b_3_0 + b_2_1·c_4_32·b_1_04 + b_2_12·c_4_32·b_1_02
         + c_4_33·b_1_02, an element of degree 14
    4. b_1_06 + b_6_0, an element of degree 6
  • A Duflot regular sequence is given by c_4_3.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 24, 36].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. b_2_12 + c_4_3, an element of degree 4
    2. b_3_14 + b_3_04 + b_1_0·b_3_12·b_5_3 + b_1_03·b_3_13 + b_1_04·b_3_1·b_5_3
         + b_1_06·b_3_02 + b_6_0·b_3_12 + b_6_02 + b_2_1·b_1_05·b_5_3
         + b_2_1·b_1_07·b_3_1 + b_2_12·b_1_02·b_3_12 + b_2_12·b_1_02·b_3_02
         + b_2_13·b_1_0·b_5_3 + b_2_13·b_1_03·b_3_1 + b_2_13·b_6_0 + b_2_14·b_1_04
         + c_4_3·b_3_1·b_5_3 + c_4_3·b_1_02·b_3_12 + c_4_3·b_1_02·b_3_02
         + c_4_3·b_1_03·b_5_3 + c_4_3·b_1_05·b_3_1 + c_4_3·b_1_05·b_3_0
         + b_2_1·c_4_3·b_3_12 + b_2_1·c_4_3·b_1_0·b_5_3 + b_2_1·c_4_3·b_1_03·b_3_0
         + b_2_1·c_4_3·b_1_06 + b_2_12·c_4_3·b_1_0·b_3_0 + b_2_13·c_4_3·b_1_02
         + c_4_32·b_1_0·b_3_0 + b_2_12·c_4_32, an element of degree 12
    3. b_7_182 + b_3_13·b_5_3 + b_1_02·b_3_04 + b_1_03·b_3_12·b_5_3 + b_1_05·b_3_13
         + b_6_0·b_3_1·b_5_3 + b_2_1·b_1_0·b_3_12·b_5_3 + b_2_1·b_1_03·b_3_13
         + b_2_1·b_6_0·b_3_12 + b_2_12·b_1_04·b_3_12 + b_2_12·b_1_04·b_3_02
         + b_2_13·b_3_1·b_5_3 + b_2_13·b_1_02·b_3_12 + b_2_13·b_1_03·b_5_3
         + b_2_13·b_1_05·b_3_1 + b_2_14·b_6_0 + c_4_3·b_1_0·b_3_13 + c_4_3·b_1_0·b_3_03
         + c_4_3·b_1_04·b_3_02 + c_4_3·b_1_05·b_5_3 + c_4_3·b_1_07·b_3_1
         + b_2_1·c_4_3·b_1_03·b_5_3 + b_2_1·c_4_3·b_1_05·b_3_0 + b_2_12·c_4_3·b_3_12
         + b_2_12·c_4_3·b_1_0·b_5_3 + b_2_12·c_4_3·b_1_03·b_3_1 + b_2_12·c_4_3·b_6_0
         + b_2_13·c_4_3·b_1_04 + c_4_32·b_3_02 + c_4_32·b_1_03·b_3_1 + c_4_32·b_1_06
         + b_2_1·c_4_32·b_1_0·b_3_0 + b_2_1·c_4_32·b_1_04 + b_2_12·c_4_32·b_1_02
         + c_4_33·b_1_02, an element of degree 14
    4. b_1_06 + b_6_0, an element of degree 6
  • We found that there exists some HSOP over a finite extension field, in degrees 4,12,6,7.

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(384,5602); GF(2))

  1. b_1_0 → b_1_1
  2. b_2_1 → b_2_4
  3. b_3_1 → b_3_0 + b_1_0·b_1_12 + b_1_02·b_1_1
  4. b_3_0 → b_3_1 + b_1_0·b_1_12 + b_1_02·b_1_1
  5. c_4_3 → b_1_02·b_1_12 + b_1_04 + b_2_4·b_1_0·b_1_1 + b_2_4·b_1_02 + b_2_3·b_2_4 + b_2_32
       + c_4_15
  6. b_5_3 → b_1_0·b_1_14 + b_1_02·b_3_0 + b_1_04·b_1_1 + b_2_4·b_1_0·b_1_12
       + b_2_4·b_1_02·b_1_1 + b_2_3·b_3_0 + b_2_3·b_2_4·b_1_0 + c_4_15·b_1_0
  7. b_6_0 → b_3_92 + b_2_3·b_1_0·b_3_0 + b_2_32·b_1_02 + b_2_3·c_4_15
  8. b_7_18 → c_4_15·b_3_9

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

  1. b_1_0 → 0, an element of degree 1
  2. b_2_1 → 0, an element of degree 2
  3. b_3_1 → 0, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_3 → c_1_04, an element of degree 4
  6. b_5_3 → 0, an element of degree 5
  7. b_6_0 → 0, an element of degree 6
  8. b_7_18 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. b_1_0 → 0, an element of degree 1
  2. b_2_1 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  3. b_3_1 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_3 → c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_1·c_1_22
       + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12
       + c_1_04, an element of degree 4
  6. b_5_3 → c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  7. b_6_0 → c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_24
       + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22, an element of degree 6
  8. b_7_18 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. b_1_0 → 0, an element of degree 1
  2. b_2_1 → 0, an element of degree 2
  3. b_3_1 → 0, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_3 → c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2
       + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_04, an element of degree 4
  6. b_5_3 → 0, an element of degree 5
  7. b_6_0 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_24 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22
       + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
  8. b_7_18 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_0 → c_1_2, an element of degree 1
  2. b_2_1 → c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_3 + c_1_0·c_1_2, an element of degree 2
  3. b_3_1 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32
       + c_1_02·c_1_3, an element of degree 3
  4. b_3_0 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_2·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_2·c_1_3
       + c_1_02·c_1_2, an element of degree 3
  5. c_4_3 → c_1_34 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_33 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_1·c_1_2·c_1_3
       + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_3 + c_1_02·c_1_12
       + c_1_03·c_1_3 + c_1_03·c_1_2 + c_1_04, an element of degree 4
  6. b_5_3 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_1·c_1_2·c_1_33
       + c_1_1·c_1_22·c_1_32 + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32
       + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_34 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_23·c_1_3
       + c_1_0·c_1_1·c_1_2·c_1_32 + c_1_0·c_1_12·c_1_2·c_1_3 + c_1_02·c_1_33
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_1·c_1_32 + c_1_02·c_1_12·c_1_3
       + c_1_03·c_1_32 + c_1_03·c_1_2·c_1_3 + c_1_04·c_1_3, an element of degree 5
  7. b_6_0 → 0, an element of degree 6
  8. b_7_18 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_0 → c_1_2, an element of degree 1
  2. b_2_1 → c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_2, an element of degree 2
  3. b_3_1 → 0, an element of degree 3
  4. b_3_0 → c_1_1·c_1_32 + c_1_1·c_1_2·c_1_3 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_02·c_1_2, an element of degree 3
  5. c_4_3 → c_1_0·c_1_1·c_1_32 + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_3
       + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_3
       + c_1_02·c_1_12 + c_1_03·c_1_2 + c_1_04, an element of degree 4
  6. b_5_3 → 0, an element of degree 5
  7. b_6_0 → 0, an element of degree 6
  8. b_7_18 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_0 → 0, an element of degree 1
  2. b_2_1 → c_1_1·c_1_3 + c_1_1·c_1_2 + c_1_12 + c_1_0·c_1_2, an element of degree 2
  3. b_3_1 → c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_22
       + c_1_02·c_1_2, an element of degree 3
  4. b_3_0 → 0, an element of degree 3
  5. c_4_3 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_1·c_1_23
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_0·c_1_1·c_1_32 + c_1_0·c_1_1·c_1_2·c_1_3
       + c_1_0·c_1_12·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_3
       + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_03·c_1_2 + c_1_04, an element of degree 4
  6. b_5_3 → c_1_1·c_1_34 + c_1_1·c_1_24 + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32
       + c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23 + c_1_0·c_1_24
       + c_1_0·c_1_1·c_1_2·c_1_32 + c_1_0·c_1_1·c_1_22·c_1_3 + c_1_0·c_1_12·c_1_2·c_1_3
       + c_1_02·c_1_23 + c_1_02·c_1_1·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_03·c_1_22 + c_1_04·c_1_2, an element of degree 5
  7. b_6_0 → c_1_22·c_1_34 + c_1_24·c_1_32 + c_1_1·c_1_2·c_1_34 + c_1_1·c_1_22·c_1_33
       + c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3 + c_1_12·c_1_2·c_1_33
       + c_1_12·c_1_23·c_1_3 + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_24·c_1_3
       + c_1_0·c_1_1·c_1_34 + c_1_0·c_1_1·c_1_22·c_1_32 + c_1_0·c_1_12·c_1_33
       + c_1_0·c_1_12·c_1_2·c_1_32 + c_1_02·c_1_34 + c_1_02·c_1_23·c_1_3
       + c_1_02·c_1_1·c_1_33 + c_1_02·c_1_1·c_1_22·c_1_3 + c_1_02·c_1_12·c_1_32
       + c_1_02·c_1_12·c_1_2·c_1_3 + c_1_03·c_1_2·c_1_32 + c_1_03·c_1_22·c_1_3
       + c_1_04·c_1_32 + c_1_04·c_1_2·c_1_3, an element of degree 6
  8. b_7_18 → 0, an element of degree 7

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