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David Green

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Mod-2-Cohomology of SymmetricGroup(7), a group of order 5040

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• SymmetricGroup(7) is a group of order 5040.
• The group order factors as 2⁴ · 3² · 5 · 7.
• The group is defined by Group([(1,2,3,4,5,6,7),(1,2)]).
• It is non-abelian.
• It has 2-Rank 3.
• The centre of a Sylow 2-subgroup has rank 2.
• Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  ( − 1)·(1  −  t  +  t2)

  ( − 1  +  t)3 · (1  +  t  +  t2)

• The a-invariants are -∞,-∞,-∞,-3. 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, -3].

About the group

Ring generators

Ring relations

Completion information

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Ring generators

The cohomology ring has 4 minimal generators of maximal degree 3:

1. c_1_0, a Duflot element of degree 1
2. c_2_0, a Duflot element of degree 2
3. b_3_3, an element of degree 3
4. c_3_2, a Duflot element of degree 3

About the group

Ring generators

Ring relations

Completion information

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Ring relations

There is one minimal relation of degree 6:

1. b_3_3² + b_3_3·c_3_2 + c_2_0·c_1_0·b_3_3

About the group

Ring generators

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Completion information

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Data used for the Hilbert-Poincaré test

• We proved completion in degree 6 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_1_0, an element of degree 1
  2. c_2_0, an element of degree 2
  3. c_3_2, an element of degree 3
• A Duflot regular sequence is given by c_1_0, c_2_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(D8xC2; GF(2))

1. c_1_0 → b_1_0 + c_1_2
2. c_2_0 → b_1_1² + b_1_1·c_1_2 + c_2_5 + c_1_2²
3. b_3_3 → c_2_5·b_1_0
4. c_3_2 → c_2_5·b_1_1 + b_1_0·c_1_2² + c_2_5·c_1_2 + c_1_2³

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

1. c_1_0 → c_1_0, an element of degree 1
2. c_2_0 → c_1_1² + c_1_0², an element of degree 2
3. b_3_3 → 0, an element of degree 3
4. c_3_2 → c_1_0·c_1_1² + c_1_0³, an element of degree 3

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

1. c_1_0 → c_1_2 + c_1_0, an element of degree 1
2. c_2_0 → c_1_1·c_1_2 + c_1_1² + c_1_0², an element of degree 2
3. b_3_3 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
4. c_3_2 → c_1_0·c_1_1·c_1_2 + c_1_0·c_1_1² + c_1_0²·c_1_2 + c_1_0³, an element of degree 3

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

1. c_1_0 → c_1_0, an element of degree 1
2. c_2_0 → c_1_2² + c_1_1·c_1_2 + c_1_1² + c_1_0·c_1_2 + c_1_0², an element of degree 2
3. b_3_3 → 0, an element of degree 3
4. c_3_2 → c_1_1·c_1_2² + c_1_1²·c_1_2 + c_1_0·c_1_1·c_1_2 + c_1_0·c_1_1² + c_1_0³, an element of degree 3

About the group

Ring generators

Ring relations

Completion information

Restriction maps