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Mod-2-Cohomology of SymmetricGroup(7), a group of order 5040
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
The computation was based on 9 stability conditions for H*(D8xC2; GF(2)).
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].
Ring generators
The cohomology ring has 4 minimal generators of maximal degree 3:
- c_1_0, a Duflot element of degree 1
- c_2_0, a Duflot element of degree 2
- b_3_3, an element of degree 3
- c_3_2, a Duflot element of degree 3
Ring relations
There is one minimal relation of degree 6:
- b_3_32 + b_3_3·c_3_2 + c_2_0·c_1_0·b_3_3
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:
- c_1_0, an element of degree 1
- c_2_0, an element of degree 2
- 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].
Restriction maps
Expressing the generators as elements of H*(D8xC2; GF(2))
- c_1_0 → b_1_0 + c_1_2
- c_2_0 → b_1_12 + b_1_1·c_1_2 + c_2_5 + c_1_22
- b_3_3 → c_2_5·b_1_0
- 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
- c_1_0 → c_1_0, an element of degree 1
- c_2_0 → c_1_12 + c_1_02, an element of degree 2
- b_3_3 → 0, an element of degree 3
- 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
- c_1_0 → c_1_2 + c_1_0, an element of degree 1
- c_2_0 → c_1_1·c_1_2 + c_1_12 + c_1_02, an element of degree 2
- b_3_3 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- 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
- c_1_0 → c_1_0, an element of degree 1
- 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
- b_3_3 → 0, an element of degree 3
- 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
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